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Theorem List for Metamath Proof Explorer - 26101-26200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempntlemb 26101* Lemma for pnt 26118. Unpack all the lower bounds contained in 𝑊, in the form they will be used. For comparison with Equation 10.6.27 of [Shapiro], p. 434, 𝑍 is x. (Contributed by Mario Carneiro, 13-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))    &   (𝜑𝑍 ∈ (𝑊[,)+∞))       (𝜑 → (𝑍 ∈ ℝ+ ∧ (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)) ∧ ((4 / (𝐿 · 𝐸)) ≤ (√‘𝑍) ∧ (((log‘𝑋) / (log‘𝐾)) + 2) ≤ (((log‘𝑍) / (log‘𝐾)) / 4) ∧ ((𝑈 · 3) + 𝐶) ≤ (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))))
 
Theorempntlemg 26102* Lemma for pnt 26118. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434, 𝑀 is j^* and 𝑁 is ĵ. (Contributed by Mario Carneiro, 13-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))    &   (𝜑𝑍 ∈ (𝑊[,)+∞))    &   𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)    &   𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))       (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ𝑀) ∧ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁𝑀)))
 
Theorempntlemh 26103* Lemma for pnt 26118. Bounds on the subintervals in the induction. (Contributed by Mario Carneiro, 13-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))    &   (𝜑𝑍 ∈ (𝑊[,)+∞))    &   𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)    &   𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))       ((𝜑𝐽 ∈ (𝑀...𝑁)) → (𝑋 < (𝐾𝐽) ∧ (𝐾𝐽) ≤ (√‘𝑍)))
 
Theorempntlemn 26104* Lemma for pnt 26118. The "naive" base bound, which we will slightly improve. (Contributed by Mario Carneiro, 13-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))    &   (𝜑𝑍 ∈ (𝑊[,)+∞))    &   𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)    &   𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))    &   (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑈)       ((𝜑 ∧ (𝐽 ∈ ℕ ∧ 𝐽 ≤ (𝑍 / 𝑌))) → 0 ≤ (((𝑈 / 𝐽) − (abs‘((𝑅‘(𝑍 / 𝐽)) / 𝑍))) · (log‘𝐽)))
 
Theorempntlemq 26105* Lemma for pntlemj 26107. (Contributed by Mario Carneiro, 7-Jun-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))    &   (𝜑𝑍 ∈ (𝑊[,)+∞))    &   𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)    &   𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))    &   (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑈)    &   (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))    &   𝑂 = (((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝐽))))    &   (𝜑𝑉 ∈ ℝ+)    &   (𝜑 → (((𝐾𝐽) < 𝑉 ∧ ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾 · (𝐾𝐽))) ∧ ∀𝑢 ∈ (𝑉[,]((1 + (𝐿 · 𝐸)) · 𝑉))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))    &   (𝜑𝐽 ∈ (𝑀..^𝑁))    &   𝐼 = (((⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) + 1)...(⌊‘(𝑍 / 𝑉)))       (𝜑𝐼𝑂)
 
Theorempntlemr 26106* Lemma for pntlemj 26107. (Contributed by Mario Carneiro, 7-Jun-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))    &   (𝜑𝑍 ∈ (𝑊[,)+∞))    &   𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)    &   𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))    &   (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑈)    &   (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))    &   𝑂 = (((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝐽))))    &   (𝜑𝑉 ∈ ℝ+)    &   (𝜑 → (((𝐾𝐽) < 𝑉 ∧ ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾 · (𝐾𝐽))) ∧ ∀𝑢 ∈ (𝑉[,]((1 + (𝐿 · 𝐸)) · 𝑉))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))    &   (𝜑𝐽 ∈ (𝑀..^𝑁))    &   𝐼 = (((⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) + 1)...(⌊‘(𝑍 / 𝑉)))       (𝜑 → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ≤ ((♯‘𝐼) · ((𝑈𝐸) · ((log‘(𝑍 / 𝑉)) / (𝑍 / 𝑉)))))
 
Theorempntlemj 26107* Lemma for pnt 26118. The induction step. Using pntibnd 26097, we find an interval in 𝐾𝐽...𝐾↑(𝐽 + 1) which is sufficiently large and has a much smaller value, 𝑅(𝑧) / 𝑧𝐸 (instead of our original bound 𝑅(𝑧) / 𝑧𝑈). (Contributed by Mario Carneiro, 13-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))    &   (𝜑𝑍 ∈ (𝑊[,)+∞))    &   𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)    &   𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))    &   (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑈)    &   (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))    &   𝑂 = (((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝐽))))    &   (𝜑𝑉 ∈ ℝ+)    &   (𝜑 → (((𝐾𝐽) < 𝑉 ∧ ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾 · (𝐾𝐽))) ∧ ∀𝑢 ∈ (𝑉[,]((1 + (𝐿 · 𝐸)) · 𝑉))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))    &   (𝜑𝐽 ∈ (𝑀..^𝑁))    &   𝐼 = (((⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) + 1)...(⌊‘(𝑍 / 𝑉)))       (𝜑 → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ≤ Σ𝑛𝑂 (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
 
Theorempntlemi 26108* Lemma for pnt 26118. Eliminate some assumptions from pntlemj 26107. (Contributed by Mario Carneiro, 13-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))    &   (𝜑𝑍 ∈ (𝑊[,)+∞))    &   𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)    &   𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))    &   (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑈)    &   (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))    &   𝑂 = (((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝐽))))       ((𝜑𝐽 ∈ (𝑀..^𝑁)) → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ≤ Σ𝑛𝑂 (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
 
Theorempntlemf 26109* Lemma for pnt 26118. Add up the pieces in pntlemi 26108 to get an estimate slightly better than the naive lower bound 0. (Contributed by Mario Carneiro, 13-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))    &   (𝜑𝑍 ∈ (𝑊[,)+∞))    &   𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)    &   𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))    &   (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑈)    &   (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))       (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
 
Theorempntlemk 26110* Lemma for pnt 26118. Evaluate the naive part of the estimate. (Contributed by Mario Carneiro, 14-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))    &   (𝜑𝑍 ∈ (𝑊[,)+∞))    &   𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)    &   𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))    &   (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑈)    &   (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))       (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) ≤ ((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)))
 
Theorempntlemo 26111* Lemma for pnt 26118. Combine all the estimates to establish a smaller eventual bound on 𝑅(𝑍) / 𝑍. (Contributed by Mario Carneiro, 14-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))    &   (𝜑𝑍 ∈ (𝑊[,)+∞))    &   𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)    &   𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))    &   (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑈)    &   (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))    &   (𝜑 → ∀𝑧 ∈ (1(,)+∞)((((abs‘(𝑅𝑧)) · (log‘𝑧)) − ((2 / (log‘𝑧)) · Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)))) / 𝑧) ≤ 𝐶)       (𝜑 → (abs‘((𝑅𝑍) / 𝑍)) ≤ (𝑈 − (𝐹 · (𝑈↑3))))
 
Theorempntleme 26112* Lemma for pnt 26118. Package up pntlemo 26111 in quantifiers. (Contributed by Mario Carneiro, 14-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))    &   (𝜑𝐶 ∈ ℝ+)    &   𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))    &   (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑈)    &   (𝜑 → ∀𝑘 ∈ (𝐾[,)+∞)∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝑘 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))    &   (𝜑 → ∀𝑧 ∈ (1(,)+∞)((((abs‘(𝑅𝑧)) · (log‘𝑧)) − ((2 / (log‘𝑧)) · Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)))) / 𝑧) ≤ 𝐶)       (𝜑 → ∃𝑤 ∈ ℝ+𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅𝑣) / 𝑣)) ≤ (𝑈 − (𝐹 · (𝑈↑3))))
 
Theorempntlem3 26113* Lemma for pnt 26118. Equation 10.6.35 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 8-Apr-2016.) (Proof shortened by AV, 27-Sep-2020.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑 → ∀𝑥 ∈ ℝ+ (abs‘((𝑅𝑥) / 𝑥)) ≤ 𝐴)    &   𝑇 = {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ+𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑡}    &   (𝜑𝐶 ∈ ℝ+)    &   ((𝜑𝑢𝑇) → (𝑢 − (𝐶 · (𝑢↑3))) ∈ 𝑇)       (𝜑 → (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ⇝𝑟 1)
 
Theorempntlemp 26114* Lemma for pnt 26118. Wrapping up more quantifiers. (Contributed by Mario Carneiro, 14-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑 → ∀𝑥 ∈ ℝ+ (abs‘((𝑅𝑥) / 𝑥)) ≤ 𝐴)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑 → ∀𝑒 ∈ (0(,)1)∃𝑥 ∈ ℝ+𝑘 ∈ ((exp‘(𝐵 / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝑒)) · 𝑧) < (𝑘 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝑒)) · 𝑧))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝑒))    &   (𝜑𝑈 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   𝐸 = (𝑈 / 𝐷)    &   𝐾 = (exp‘(𝐵 / 𝐸))    &   (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))    &   (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑈)       (𝜑 → ∃𝑤 ∈ ℝ+𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅𝑣) / 𝑣)) ≤ (𝑈 − (𝐹 · (𝑈↑3))))
 
Theorempntleml 26115* Lemma for pnt 26118. Equation 10.6.35 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 14-Apr-2016.)
𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑 → ∀𝑥 ∈ ℝ+ (abs‘((𝑅𝑥) / 𝑥)) ≤ 𝐴)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐿 ∈ (0(,)1))    &   𝐷 = (𝐴 + 1)    &   𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))    &   (𝜑 → ∀𝑒 ∈ (0(,)1)∃𝑥 ∈ ℝ+𝑘 ∈ ((exp‘(𝐵 / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝑒)) · 𝑧) < (𝑘 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝑒)) · 𝑧))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝑒))       (𝜑 → (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ⇝𝑟 1)
 
Theorempnt3 26116 The Prime Number Theorem, version 3: the second Chebyshev function tends asymptotically to 𝑥. (Contributed by Mario Carneiro, 1-Jun-2016.)
(𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ⇝𝑟 1
 
Theorempnt2 26117 The Prime Number Theorem, version 2: the first Chebyshev function tends asymptotically to 𝑥. (Contributed by Mario Carneiro, 1-Jun-2016.)
(𝑥 ∈ ℝ+ ↦ ((θ‘𝑥) / 𝑥)) ⇝𝑟 1
 
Theorempnt 26118 The Prime Number Theorem: the number of prime numbers less than 𝑥 tends asymptotically to 𝑥 / log(𝑥) as 𝑥 goes to infinity. This is Metamath 100 proof #5. (Contributed by Mario Carneiro, 1-Jun-2016.)
(𝑥 ∈ (1(,)+∞) ↦ ((π𝑥) / (𝑥 / (log‘𝑥)))) ⇝𝑟 1
 
14.4.14  Ostrowski's theorem
 
Theoremabvcxp 26119* Raising an absolute value to a power less than one yields another absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &   𝐺 = (𝑥𝐵 ↦ ((𝐹𝑥)↑𝑐𝑆))       ((𝐹𝐴𝑆 ∈ (0(,]1)) → 𝐺𝐴)
 
Theorempadicfval 26120* Value of the p-adic absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥)))))       (𝑃 ∈ ℙ → (𝐽𝑃) = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑃↑-(𝑃 pCnt 𝑥)))))
 
Theorempadicval 26121* Value of the p-adic absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥)))))       ((𝑃 ∈ ℙ ∧ 𝑋 ∈ ℚ) → ((𝐽𝑃)‘𝑋) = if(𝑋 = 0, 0, (𝑃↑-(𝑃 pCnt 𝑋))))
 
Theoremostth2lem1 26122* Lemma for ostth2 26141, although it is just a simple statement about exponentials which does not involve any specifics of ostth2 26141. If a power is upper bounded by a linear term, the exponent must be less than one. Or in big-O notation, 𝑛𝑜(𝐴𝑛) for any 1 < 𝐴. (Contributed by Mario Carneiro, 10-Sep-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   ((𝜑𝑛 ∈ ℕ) → (𝐴𝑛) ≤ (𝑛 · 𝐵))       (𝜑𝐴 ≤ 1)
 
Theoremqrngbas 26123 The base set of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝑄 = (ℂflds ℚ)       ℚ = (Base‘𝑄)
 
Theoremqdrng 26124 The rationals form a division ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝑄 = (ℂflds ℚ)       𝑄 ∈ DivRing
 
Theoremqrng0 26125 The zero element of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝑄 = (ℂflds ℚ)       0 = (0g𝑄)
 
Theoremqrng1 26126 The unit element of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝑄 = (ℂflds ℚ)       1 = (1r𝑄)
 
Theoremqrngneg 26127 The additive inverse in the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝑄 = (ℂflds ℚ)       (𝑋 ∈ ℚ → ((invg𝑄)‘𝑋) = -𝑋)
 
Theoremqrngdiv 26128 The division operation in the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝑄 = (ℂflds ℚ)       ((𝑋 ∈ ℚ ∧ 𝑌 ∈ ℚ ∧ 𝑌 ≠ 0) → (𝑋(/r𝑄)𝑌) = (𝑋 / 𝑌))
 
Theoremqabvle 26129 By using induction on 𝑁, we show a long-range inequality coming from the triangle inequality. (Contributed by Mario Carneiro, 10-Sep-2014.)
𝑄 = (ℂflds ℚ)    &   𝐴 = (AbsVal‘𝑄)       ((𝐹𝐴𝑁 ∈ ℕ0) → (𝐹𝑁) ≤ 𝑁)
 
Theoremqabvexp 26130 Induct the product rule abvmul 19531 to find the absolute value of a power. (Contributed by Mario Carneiro, 10-Sep-2014.)
𝑄 = (ℂflds ℚ)    &   𝐴 = (AbsVal‘𝑄)       ((𝐹𝐴𝑀 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (𝐹‘(𝑀𝑁)) = ((𝐹𝑀)↑𝑁))
 
Theoremostthlem1 26131* Lemma for ostth 26143. If two absolute values agree on the positive integers greater than one, then they agree for all rational numbers and thus are equal as functions. (Contributed by Mario Carneiro, 9-Sep-2014.)
𝑄 = (ℂflds ℚ)    &   𝐴 = (AbsVal‘𝑄)    &   (𝜑𝐹𝐴)    &   (𝜑𝐺𝐴)    &   ((𝜑𝑛 ∈ (ℤ‘2)) → (𝐹𝑛) = (𝐺𝑛))       (𝜑𝐹 = 𝐺)
 
Theoremostthlem2 26132* Lemma for ostth 26143. Refine ostthlem1 26131 so that it is sufficient to only show equality on the primes. (Contributed by Mario Carneiro, 9-Sep-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)
𝑄 = (ℂflds ℚ)    &   𝐴 = (AbsVal‘𝑄)    &   (𝜑𝐹𝐴)    &   (𝜑𝐺𝐴)    &   ((𝜑𝑝 ∈ ℙ) → (𝐹𝑝) = (𝐺𝑝))       (𝜑𝐹 = 𝐺)
 
Theoremqabsabv 26133 The regular absolute value function on the rationals is in fact an absolute value under our definition. (Contributed by Mario Carneiro, 9-Sep-2014.)
𝑄 = (ℂflds ℚ)    &   𝐴 = (AbsVal‘𝑄)       (abs ↾ ℚ) ∈ 𝐴
 
Theorempadicabv 26134* The p-adic absolute value (with arbitrary base) is an absolute value. (Contributed by Mario Carneiro, 9-Sep-2014.)
𝑄 = (ℂflds ℚ)    &   𝐴 = (AbsVal‘𝑄)    &   𝐹 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑁↑(𝑃 pCnt 𝑥))))       ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → 𝐹𝐴)
 
Theorempadicabvf 26135* The p-adic absolute value is an absolute value. (Contributed by Mario Carneiro, 9-Sep-2014.)
𝑄 = (ℂflds ℚ)    &   𝐴 = (AbsVal‘𝑄)    &   𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥)))))       𝐽:ℙ⟶𝐴
 
Theorempadicabvcxp 26136* All positive powers of the p-adic absolute value are absolute values. (Contributed by Mario Carneiro, 9-Sep-2014.)
𝑄 = (ℂflds ℚ)    &   𝐴 = (AbsVal‘𝑄)    &   𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥)))))       ((𝑃 ∈ ℙ ∧ 𝑅 ∈ ℝ+) → (𝑦 ∈ ℚ ↦ (((𝐽𝑃)‘𝑦)↑𝑐𝑅)) ∈ 𝐴)
 
Theoremostth1 26137* - Lemma for ostth 26143: trivial case. (Not that the proof is trivial, but that we are proving that the function is trivial.) If 𝐹 is equal to 1 on the primes, then by complete induction and the multiplicative property abvmul 19531 of the absolute value, 𝐹 is equal to 1 on all the integers, and ostthlem1 26131 extends this to the other rational numbers. (Contributed by Mario Carneiro, 10-Sep-2014.)
𝑄 = (ℂflds ℚ)    &   𝐴 = (AbsVal‘𝑄)    &   𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥)))))    &   𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1))    &   (𝜑𝐹𝐴)    &   (𝜑 → ∀𝑛 ∈ ℕ ¬ 1 < (𝐹𝑛))    &   (𝜑 → ∀𝑛 ∈ ℙ ¬ (𝐹𝑛) < 1)       (𝜑𝐹 = 𝐾)
 
Theoremostth2lem2 26138* Lemma for ostth2 26141. (Contributed by Mario Carneiro, 10-Sep-2014.)
𝑄 = (ℂflds ℚ)    &   𝐴 = (AbsVal‘𝑄)    &   𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥)))))    &   𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1))    &   (𝜑𝐹𝐴)    &   (𝜑𝑁 ∈ (ℤ‘2))    &   (𝜑 → 1 < (𝐹𝑁))    &   𝑅 = ((log‘(𝐹𝑁)) / (log‘𝑁))    &   (𝜑𝑀 ∈ (ℤ‘2))    &   𝑆 = ((log‘(𝐹𝑀)) / (log‘𝑀))    &   𝑇 = if((𝐹𝑀) ≤ 1, 1, (𝐹𝑀))       ((𝜑𝑋 ∈ ℕ0𝑌 ∈ (0...((𝑀𝑋) − 1))) → (𝐹𝑌) ≤ ((𝑀 · 𝑋) · (𝑇𝑋)))
 
Theoremostth2lem3 26139* Lemma for ostth2 26141. (Contributed by Mario Carneiro, 10-Sep-2014.)
𝑄 = (ℂflds ℚ)    &   𝐴 = (AbsVal‘𝑄)    &   𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥)))))    &   𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1))    &   (𝜑𝐹𝐴)    &   (𝜑𝑁 ∈ (ℤ‘2))    &   (𝜑 → 1 < (𝐹𝑁))    &   𝑅 = ((log‘(𝐹𝑁)) / (log‘𝑁))    &   (𝜑𝑀 ∈ (ℤ‘2))    &   𝑆 = ((log‘(𝐹𝑀)) / (log‘𝑀))    &   𝑇 = if((𝐹𝑀) ≤ 1, 1, (𝐹𝑀))    &   𝑈 = ((log‘𝑁) / (log‘𝑀))       ((𝜑𝑋 ∈ ℕ) → (((𝐹𝑁) / (𝑇𝑐𝑈))↑𝑋) ≤ (𝑋 · ((𝑀 · 𝑇) · (𝑈 + 1))))
 
Theoremostth2lem4 26140* Lemma for ostth2 26141. (Contributed by Mario Carneiro, 10-Sep-2014.)
𝑄 = (ℂflds ℚ)    &   𝐴 = (AbsVal‘𝑄)    &   𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥)))))    &   𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1))    &   (𝜑𝐹𝐴)    &   (𝜑𝑁 ∈ (ℤ‘2))    &   (𝜑 → 1 < (𝐹𝑁))    &   𝑅 = ((log‘(𝐹𝑁)) / (log‘𝑁))    &   (𝜑𝑀 ∈ (ℤ‘2))    &   𝑆 = ((log‘(𝐹𝑀)) / (log‘𝑀))    &   𝑇 = if((𝐹𝑀) ≤ 1, 1, (𝐹𝑀))    &   𝑈 = ((log‘𝑁) / (log‘𝑀))       (𝜑 → (1 < (𝐹𝑀) ∧ 𝑅𝑆))
 
Theoremostth2 26141* - Lemma for ostth 26143: regular case. (Contributed by Mario Carneiro, 10-Sep-2014.)
𝑄 = (ℂflds ℚ)    &   𝐴 = (AbsVal‘𝑄)    &   𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥)))))    &   𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1))    &   (𝜑𝐹𝐴)    &   (𝜑𝑁 ∈ (ℤ‘2))    &   (𝜑 → 1 < (𝐹𝑁))    &   𝑅 = ((log‘(𝐹𝑁)) / (log‘𝑁))       (𝜑 → ∃𝑎 ∈ (0(,]1)𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑𝑐𝑎)))
 
Theoremostth3 26142* - Lemma for ostth 26143: p-adic case. (Contributed by Mario Carneiro, 10-Sep-2014.)
𝑄 = (ℂflds ℚ)    &   𝐴 = (AbsVal‘𝑄)    &   𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥)))))    &   𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1))    &   (𝜑𝐹𝐴)    &   (𝜑 → ∀𝑛 ∈ ℕ ¬ 1 < (𝐹𝑛))    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑 → (𝐹𝑃) < 1)    &   𝑅 = -((log‘(𝐹𝑃)) / (log‘𝑃))    &   𝑆 = if((𝐹𝑃) ≤ (𝐹𝑝), (𝐹𝑝), (𝐹𝑃))       (𝜑 → ∃𝑎 ∈ ℝ+ 𝐹 = (𝑦 ∈ ℚ ↦ (((𝐽𝑃)‘𝑦)↑𝑐𝑎)))
 
Theoremostth 26143* Ostrowski's theorem, which classifies all absolute values on . Any such absolute value must either be the trivial absolute value 𝐾, a constant exponent 0 < 𝑎 ≤ 1 times the regular absolute value, or a positive exponent times the p-adic absolute value. (Contributed by Mario Carneiro, 10-Sep-2014.)
𝑄 = (ℂflds ℚ)    &   𝐴 = (AbsVal‘𝑄)    &   𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥)))))    &   𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1))       (𝐹𝐴 ↔ (𝐹 = 𝐾 ∨ ∃𝑎 ∈ (0(,]1)𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑𝑐𝑎)) ∨ ∃𝑎 ∈ ℝ+𝑔 ∈ ran 𝐽 𝐹 = (𝑦 ∈ ℚ ↦ ((𝑔𝑦)↑𝑐𝑎))))
 
PART 15  ELEMENTARY GEOMETRY

This part develops elementary geometry based on Tarski's axioms, following [Schwabhauser]. Tarski's geometry is a first-order theory with one sort, the "points". It has two primitive notions, the ternary predicate of "betweenness" and the quaternary predicate of "congruence". To adapt this theory to the framework of set.mm, and to be able to talk of *a* Tarski structure as a space satisfying the given axioms, we use the following definition, stated informally:

A Tarski structure 𝑓 is a set (of points) (Base‘𝑓) together with functions (Itv‘𝑓) and (dist‘𝑓) on ((Base‘𝑓) × (Base‘𝑓)) satisfying certain axioms (given in the definitions df-trkg 26167 et sequentes). This allows us to treat a Tarski structure as a special kind of extensible structure (see df-struct 16475).

The translation to and from Tarski's treatment is as follows (given, again, informally).

Suppose that one is given an extensible structure 𝑓. One defines a betweenness ternary predicate Btw by positing that, for any 𝑥, 𝑦, 𝑧 ∈ (Base‘𝑓), one has "Btw 𝑥𝑦𝑧 " if and only if 𝑦𝑥(Itv‘𝑓)𝑧, and a congruence quaternary predicate Congr by positing that, for any 𝑥, 𝑦, 𝑧, 𝑡 ∈ (Base‘𝑓), one has "Congr 𝑥𝑦𝑧𝑡 " if and only if 𝑥(dist‘𝑓)𝑦 = 𝑧(dist‘𝑓)𝑡. It is easy to check that if 𝑓 satisfies our Tarski axioms, then Btw and Congr satisfy Tarski's Tarski axioms when (Base‘𝑓) is interpreted as the universe of discourse.

Conversely, suppose that one is given a set 𝑎, a ternary predicate Btw, and a quaternary predicate Congr. One defines the extensible structure 𝑓 such that (Base‘𝑓) is 𝑎, and (Itv‘𝑓) is the function which associates with each 𝑥, 𝑦⟩ ∈ (𝑎 × 𝑎) the set of points 𝑧𝑎 such that "Btw 𝑥𝑧𝑦", and (dist‘𝑓) is the function which associates with each 𝑥, 𝑦⟩ ∈ (𝑎 × 𝑎) the set of ordered pairs 𝑧, 𝑡⟩ ∈ (𝑎 × 𝑎) such that "Congr 𝑥𝑦𝑧𝑡". It is easy to check that if Btw and Congr satisfy Tarski's Tarski axioms when 𝑎 is interpreted as the universe of discourse, then 𝑓 satisfies our Tarski axioms.

We intentionally choose to represent congruence (without loss of generality) as 𝑥(dist‘𝑓)𝑦 = 𝑧(dist‘𝑓)𝑡 instead of "Congr 𝑥𝑦𝑧𝑡", as it is more convenient. It is always possible to define dist for any particular geometry to produce equal results when conguence is desired, and in many cases there is an obvious interpretation of "distance" between two points that can be useful in other situations. Encoding congruence as an equality of distances makes it easier to use these theorems in cases where there is a preferred distance function. We prove that representing a congruence relationship using a distance in the form 𝑥(dist‘𝑓)𝑦 = 𝑧(dist‘𝑓)𝑡 causes no loss of generality in tgjustc1 26189 and tgjustc2 26190, which in turn are supported by tgjustf 26187 and tgjustr 26188.

A similar representation of congruence (using a "distance" function) is used in Axiom A1 of [Beeson2016] p. 5, which discusses how a large number of formalized proofs were found in Tarskian Geometry using OTTER. Their detailed proofs in Tarski Geometry, along with other information, are available at http://www.michaelbeeson.com/research/FormalTarski/ 26188.

Most theorems are in deduction form, as this is a very general, simple, and convenient format to use in Metamath. An assertion in deduction form can be easily converted into an assertion in inference form (removing the antecedents 𝜑) by insert a ⊤ → in each hypothesis, using a1i 11, then using mptru 1535 to remove the final ⊤ → prefix. In some cases we represent, without loss of generality, an implication antecedent in [Schwabhauser] as a hypothesis. The implication can be retrieved from the by using simpr 485, the theorem as stated, and ex 413.

For descriptions of individual axioms, we refer to the specific definitions below. A particular feature of Tarski's axioms is modularity, so by using various subsets of the set of axioms, we can define the classes of "absolute dimensionless Tarski structures" (df-trkg 26167), of "Euclidean dimensionless Tarski structures" (df-trkge 26165) and of "Tarski structures of dimension no less than N" (df-trkgld 26166).

In this system, angles are not a primitive notion, but instead a derived notion (see df-cgra 26522 and iscgra 26523). To maintain its simplicity, in this system congruence between shapes (a finite sequence of points) is the case where corresponding segments between all corresponding points are congruent. This includes triangles (a shape of 3 distinct points). Note that this definition has no direct regard for angles. For more details and rationale, see df-cgrg 26225.

The first section is devoted to the definitions of these various structures. The second section ("Tarskian geometry") develops the synthetic treatment of geometry. The remaining sections prove that real Euclidean spaces and complex Hilbert spaces, with intended interpretations, are Euclidean Tarski structures.

Most of the work in this part is due to Thierry Arnoux, with earlier work by Mario Carneiro and Scott Fenton. See also the credits in the comment of each statement.

 
15.1  Definition and Tarski's Axioms of Geometry
 
Syntaxcstrkg 26144 Extends class notation with the class of Tarski geometries.
class TarskiG
 
Syntaxcstrkgc 26145 Extends class notation with the class of geometries fulfilling the congruence axioms.
class TarskiGC
 
Syntaxcstrkgb 26146 Extends class notation with the class of geometries fulfilling the betweenness axioms.
class TarskiGB
 
Syntaxcstrkgcb 26147 Extends class notation with the class of geometries fulfilling the congruence and betweenness axioms.
class TarskiGCB
 
Syntaxcstrkgld 26148 Extends class notation with the relation for geometries fulfilling the lower dimension axioms.
class DimTarskiG
 
Syntaxcstrkge 26149 Extends class notation with the class of geometries fulfilling Euclid's axiom.
class TarskiGE
 
Syntaxcitv 26150 Declare the syntax for the Interval (segment) index extractor.
class Itv
 
Syntaxclng 26151 Declare the syntax for the Line function.
class LineG
 
Definitiondf-itv 26152 Define the Interval (segment) index extractor for Tarski geometries. (Contributed by Thierry Arnoux, 24-Aug-2017.)
Itv = Slot 16
 
Definitiondf-lng 26153 Define the line index extractor for geometries. (Contributed by Thierry Arnoux, 27-Mar-2019.)
LineG = Slot 17
 
Theoremitvndx 26154 Index value of the Interval (segment) slot. Use ndxarg 16498. (Contributed by Thierry Arnoux, 24-Aug-2017.)
(Itv‘ndx) = 16
 
Theoremlngndx 26155 Index value of the "line" slot. Use ndxarg 16498. (Contributed by Thierry Arnoux, 27-Mar-2019.)
(LineG‘ndx) = 17
 
Theoremitvid 26156 Utility theorem: index-independent form of df-itv 26152. (Contributed by Thierry Arnoux, 24-Aug-2017.)
Itv = Slot (Itv‘ndx)
 
Theoremlngid 26157 Utility theorem: index-independent form of df-lng 26153. (Contributed by Thierry Arnoux, 27-Mar-2019.)
LineG = Slot (LineG‘ndx)
 
Theoremtrkgstr 26158 Functionality of a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017.)
𝑊 = {⟨(Base‘ndx), 𝑈⟩, ⟨(dist‘ndx), 𝐷⟩, ⟨(Itv‘ndx), 𝐼⟩}       𝑊 Struct ⟨1, 16⟩
 
Theoremtrkgbas 26159 The base set of a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017.)
𝑊 = {⟨(Base‘ndx), 𝑈⟩, ⟨(dist‘ndx), 𝐷⟩, ⟨(Itv‘ndx), 𝐼⟩}       (𝑈𝑉𝑈 = (Base‘𝑊))
 
Theoremtrkgdist 26160 The measure of a distance in a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017.)
𝑊 = {⟨(Base‘ndx), 𝑈⟩, ⟨(dist‘ndx), 𝐷⟩, ⟨(Itv‘ndx), 𝐼⟩}       (𝐷𝑉𝐷 = (dist‘𝑊))
 
Theoremtrkgitv 26161 The congruence relation in a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017.)
𝑊 = {⟨(Base‘ndx), 𝑈⟩, ⟨(dist‘ndx), 𝐷⟩, ⟨(Itv‘ndx), 𝐼⟩}       (𝐼𝑉𝐼 = (Itv‘𝑊))
 
Definitiondf-trkgc 26162* Define the class of geometries fulfilling the congruence axioms of reflexivity, identity and transitivity. These are axioms A1 to A3 of [Schwabhauser] p. 10. With our distance based notation for congruence, transitivity of congruence boils down to transitivity of equality and is already given by eqtr 2841, so it is not listed in this definition. (Contributed by Thierry Arnoux, 24-Aug-2017.)
TarskiGC = {𝑓[(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))}
 
Definitiondf-trkgb 26163* Define the class of geometries fulfilling the 3 betweenness axioms in Tarski's Axiomatization of Geometry: identity, Axiom A6 of [Schwabhauser] p. 11, axiom of Pasch, Axiom A7 of [Schwabhauser] p. 12, and continuity, Axiom A11 of [Schwabhauser] p. 13. (Contributed by Thierry Arnoux, 24-Aug-2017.)
TarskiGB = {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](∀𝑥𝑝𝑦𝑝 (𝑦 ∈ (𝑥𝑖𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝𝑢𝑝𝑣𝑝 ((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) → ∃𝑎𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑝𝑡 ∈ 𝒫 𝑝(∃𝑎𝑝𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝑖𝑦) → ∃𝑏𝑝𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝑖𝑦)))}
 
Definitiondf-trkgcb 26164* Define the class of geometries fulfilling the five segment axiom, Axiom A5 of [Schwabhauser] p. 11, and segment construction axiom, Axiom A4 of [Schwabhauser] p. 11. (Contributed by Thierry Arnoux, 14-Mar-2019.)
TarskiGCB = {𝑓[(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑][(Itv‘𝑓) / 𝑖](∀𝑥𝑝𝑦𝑝𝑧𝑝𝑢𝑝𝑎𝑝𝑏𝑝𝑐𝑝𝑣𝑝 (((𝑥𝑦𝑦 ∈ (𝑥𝑖𝑧) ∧ 𝑏 ∈ (𝑎𝑖𝑐)) ∧ (((𝑥𝑑𝑦) = (𝑎𝑑𝑏) ∧ (𝑦𝑑𝑧) = (𝑏𝑑𝑐)) ∧ ((𝑥𝑑𝑢) = (𝑎𝑑𝑣) ∧ (𝑦𝑑𝑢) = (𝑏𝑑𝑣)))) → (𝑧𝑑𝑢) = (𝑐𝑑𝑣)) ∧ ∀𝑥𝑝𝑦𝑝𝑎𝑝𝑏𝑝𝑧𝑝 (𝑦 ∈ (𝑥𝑖𝑧) ∧ (𝑦𝑑𝑧) = (𝑎𝑑𝑏)))}
 
Definitiondf-trkge 26165* Define the class of geometries fulfilling Euclid's axiom, Axiom A10 of [Schwabhauser] p. 13. (Contributed by Thierry Arnoux, 14-Mar-2019.)
TarskiGE = {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖]𝑥𝑝𝑦𝑝𝑧𝑝𝑢𝑝𝑣𝑝 ((𝑢 ∈ (𝑥𝑖𝑣) ∧ 𝑢 ∈ (𝑦𝑖𝑧) ∧ 𝑥𝑢) → ∃𝑎𝑝𝑏𝑝 (𝑦 ∈ (𝑥𝑖𝑎) ∧ 𝑧 ∈ (𝑥𝑖𝑏) ∧ 𝑣 ∈ (𝑎𝑖𝑏)))}
 
Definitiondf-trkgld 26166* Define the class of geometries fulfilling the lower dimension axiom for dimension 𝑛. For such geometries, there are three non-colinear points that are equidistant from 𝑛 − 1 distinct points. Derived from remarks in Tarski's System of Geometry, Alfred Tarski and Steven Givant, Bulletin of Symbolic Logic, Volume 5, Number 2 (1999), 175-214. (Contributed by Scott Fenton, 22-Apr-2013.) (Revised by Thierry Arnoux, 23-Nov-2019.)
DimTarskiG≥ = {⟨𝑔, 𝑛⟩ ∣ [(Base‘𝑔) / 𝑝][(dist‘𝑔) / 𝑑][(Itv‘𝑔) / 𝑖]𝑓(𝑓:(1..^𝑛)–1-1𝑝 ∧ ∃𝑥𝑝𝑦𝑝𝑧𝑝 (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1)𝑑𝑥) = ((𝑓𝑗)𝑑𝑥) ∧ ((𝑓‘1)𝑑𝑦) = ((𝑓𝑗)𝑑𝑦) ∧ ((𝑓‘1)𝑑𝑧) = ((𝑓𝑗)𝑑𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))}
 
Definitiondf-trkg 26167* Define the class of Tarski geometries. A Tarski geometry is a set of points, equipped with a betweenness relation (denoting that a point lies on a line segment between two other points) and a congruence relation (denoting equality of line segment lengths). Here, we are using the following:
  • for congruence, (𝑥 𝑦) = (𝑧 𝑤) where = (dist‘𝑊)
  • for betweenness, 𝑦 ∈ (𝑥𝐼𝑧), where 𝐼 = (Itv‘𝑊)
With this definition, the axiom A2 is actually equivalent to the transitivity of addition, eqtrd 2856.

Tarski originally had more axioms, but later reduced his list to 11:

  • A1 A kind of reflexivity for the congruence relation (TarskiGC)
  • A2 Transitivity for the congruence relation (TarskiGC)
  • A3 Identity for the congruence relation (TarskiGC)
  • A4 Axiom of segment construction (TarskiGCB)
  • A5 5-segment axiom (TarskiGCB)
  • A6 Identity for the betweenness relation (TarskiGB)
  • A7 Axiom of Pasch (TarskiGB)
  • A8 Lower dimension axiom (DimTarskiG≥‘2)
  • A9 Upper dimension axiom (V ∖ (DimTarskiG≥‘3))
  • A10 Euclid's axiom (TarskiGE)
  • A11 Axiom of continuity (TarskiGB)
Our definition is split into 5 parts:
  • congruence axioms TarskiGC (which metric spaces fulfill)
  • betweenness axioms TarskiGB
  • congruence and betweenness axioms TarskiGCB
  • upper and lower dimension axioms DimTarskiG
  • axiom of Euclid / parallel postulate TarskiGE

So our definition of a Tarskian Geometry includes the 3 axioms for the quaternary congruence relation (A1, A2, A3), the 3 axioms for the ternary betweenness relation (A6, A7, A11), and the 2 axioms of compatibility of the congruence and the betweenness relations (A4,A5).

It does not include Euclid's axiom A10, nor the 2-dimensional axioms A8 (Lower dimension axiom) and A9 (Upper dimension axiom) so the number of dimensions of the geometry it formalizes is not constrained.

Considering A2 as one of the 3 axioms for the quaternary congruence relation is somewhat conventional, because the transitivity of the congruence relation is automatically given by our choice to take the distance as this congruence relation in our definition of Tarski geometries. (Contributed by Thierry Arnoux, 24-Aug-2017.) (Revised by Thierry Arnoux, 27-Apr-2019.)

TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
 
Theoremistrkgc 26168* Property of being a Tarski geometry - congruence part. (Contributed by Thierry Arnoux, 14-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)       (𝐺 ∈ TarskiGC ↔ (𝐺 ∈ V ∧ (∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦))))
 
Theoremistrkgb 26169* Property of being a Tarski geometry - betweenness part. (Contributed by Thierry Arnoux, 14-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)       (𝐺 ∈ TarskiGB ↔ (𝐺 ∈ V ∧ (∀𝑥𝑃𝑦𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑃𝑡 ∈ 𝒫 𝑃(∃𝑎𝑃𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦)))))
 
Theoremistrkgcb 26170* Property of being a Tarski geometry - congruence and betweenness part. (Contributed by Thierry Arnoux, 14-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)       (𝐺 ∈ TarskiGCB ↔ (𝐺 ∈ V ∧ (∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑎𝑃𝑏𝑃𝑐𝑃𝑣𝑃 (((𝑥𝑦𝑦 ∈ (𝑥𝐼𝑧) ∧ 𝑏 ∈ (𝑎𝐼𝑐)) ∧ (((𝑥 𝑦) = (𝑎 𝑏) ∧ (𝑦 𝑧) = (𝑏 𝑐)) ∧ ((𝑥 𝑢) = (𝑎 𝑣) ∧ (𝑦 𝑢) = (𝑏 𝑣)))) → (𝑧 𝑢) = (𝑐 𝑣)) ∧ ∀𝑥𝑃𝑦𝑃𝑎𝑃𝑏𝑃𝑧𝑃 (𝑦 ∈ (𝑥𝐼𝑧) ∧ (𝑦 𝑧) = (𝑎 𝑏)))))
 
Theoremistrkge 26171* Property of fulfilling Euclid's axiom. (Contributed by Thierry Arnoux, 14-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)       (𝐺 ∈ TarskiGE ↔ (𝐺 ∈ V ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑣) ∧ 𝑢 ∈ (𝑦𝐼𝑧) ∧ 𝑥𝑢) → ∃𝑎𝑃𝑏𝑃 (𝑦 ∈ (𝑥𝐼𝑎) ∧ 𝑧 ∈ (𝑥𝐼𝑏) ∧ 𝑣 ∈ (𝑎𝐼𝑏)))))
 
Theoremistrkgl 26172* Building lines from the segment property. (Contributed by Thierry Arnoux, 14-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)       (𝐺 ∈ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})} ↔ (𝐺 ∈ V ∧ (LineG‘𝐺) = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})))
 
Theoremistrkgld 26173* Property of fulfilling the lower dimension 𝑁 axiom. (Contributed by Thierry Arnoux, 20-Nov-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)       ((𝐺𝑉𝑁 ∈ (ℤ‘2)) → (𝐺DimTarskiG𝑁 ↔ ∃𝑓(𝑓:(1..^𝑁)–1-1𝑃 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (∀𝑗 ∈ (2..^𝑁)(((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ∧ ((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ∧ ((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
 
Theoremistrkg2ld 26174* Property of fulfilling the lower dimension 2 axiom. (Contributed by Thierry Arnoux, 20-Nov-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)       (𝐺𝑉 → (𝐺DimTarskiG≥2 ↔ ∃𝑥𝑃𝑦𝑃𝑧𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))
 
Theoremistrkg3ld 26175* Property of fulfilling the lower dimension 3 axiom. (Contributed by Thierry Arnoux, 12-Jul-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)       (𝐺𝑉 → (𝐺DimTarskiG≥3 ↔ ∃𝑢𝑃𝑣𝑃 (𝑢𝑣 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑢 𝑥) = (𝑣 𝑥) ∧ (𝑢 𝑦) = (𝑣 𝑦) ∧ (𝑢 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
 
Theoremaxtgcgrrflx 26176 Axiom of reflexivity of congruence, Axiom A1 of [Schwabhauser] p. 10. (Contributed by Thierry Arnoux, 14-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)       (𝜑 → (𝑋 𝑌) = (𝑌 𝑋))
 
Theoremaxtgcgrid 26177 Axiom of identity of congruence, Axiom A3 of [Schwabhauser] p. 10. (Contributed by Thierry Arnoux, 14-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑 → (𝑋 𝑌) = (𝑍 𝑍))       (𝜑𝑋 = 𝑌)
 
Theoremaxtgsegcon 26178* Axiom of segment construction, Axiom A4 of [Schwabhauser] p. 11. As discussed in Axiom 4 of [Tarski1999] p. 178, "The intuitive content [is that] given any line segment 𝐴𝐵, one can construct a line segment congruent to it, starting at any point 𝑌 and going in the direction of any ray containing 𝑌. The ray is determined by the point 𝑌 and a second point 𝑋, the endpoint of the ray. The other endpoint of the line segment to be constructed is just the point 𝑧 whose existence is asserted." (Contributed by Thierry Arnoux, 15-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → ∃𝑧𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 𝑧) = (𝐴 𝐵)))
 
Theoremaxtg5seg 26179 Five segments axiom, Axiom A5 of [Schwabhauser] p. 11. Take two triangles 𝑋𝑍𝑈 and 𝐴𝐶𝑉, a point 𝑌 on 𝑋𝑍, and a point 𝐵 on 𝐴𝐶. If all corresponding line segments except for 𝑍𝑈 and 𝐶𝑉 are congruent ( i.e., 𝑋𝑌 𝐴𝐵, 𝑌𝑍 𝐵𝐶, 𝑋𝑈 𝐴𝑉, and 𝑌𝑈 𝐵𝑉), then 𝑍𝑈 and 𝐶𝑉 are also congruent. As noted in Axiom 5 of [Tarski1999] p. 178, "this axiom is similar in character to the well-known theorems of Euclidean geometry that allow one to conclude, from hypotheses about the congruence of certain corresponding sides and angles in two triangles, the congruence of other corresponding sides and angles." (Contributed by Thierry Arnoux, 14-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝑈𝑃)    &   (𝜑𝑉𝑃)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌 ∈ (𝑋𝐼𝑍))    &   (𝜑𝐵 ∈ (𝐴𝐼𝐶))    &   (𝜑 → (𝑋 𝑌) = (𝐴 𝐵))    &   (𝜑 → (𝑌 𝑍) = (𝐵 𝐶))    &   (𝜑 → (𝑋 𝑈) = (𝐴 𝑉))    &   (𝜑 → (𝑌 𝑈) = (𝐵 𝑉))       (𝜑 → (𝑍 𝑈) = (𝐶 𝑉))
 
Theoremaxtgbtwnid 26180 Identity of Betweenness. Axiom A6 of [Schwabhauser] p. 11. (Contributed by Thierry Arnoux, 15-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑌 ∈ (𝑋𝐼𝑋))       (𝜑𝑋 = 𝑌)
 
Theoremaxtgpasch 26181* Axiom of (Inner) Pasch, Axiom A7 of [Schwabhauser] p. 12. Given triangle 𝑋𝑌𝑍, point 𝑈 in segment 𝑋𝑍, and point 𝑉 in segment 𝑌𝑍, there exists a point 𝑎 on both the segment 𝑈𝑌 and the segment 𝑉𝑋. This axiom is essentially a subset of the general Pasch axiom. The general Pasch axiom asserts that on a plane "a line intersecting a triangle in one of its sides, and not intersecting any of the vertices, must intersect one of the other two sides" (per the discussion about Axiom 7 of [Tarski1999] p. 179). The (general) Pasch axiom was used implicitly by Euclid, but never stated; Moritz Pasch discovered its omission in 1882. As noted in the Metamath book, this means that the omission of Pasch's axiom from Euclid went unnoticed for 2000 years. Only the inner Pasch algorithm is included as an axiom; the "outer" form of the Pasch axiom can be proved using the inner form (see theorem 9.6 of [Schwabhauser] p. 69 and the brief discussion in axiom 7.1 of [Tarski1999] p. 180). (Contributed by Thierry Arnoux, 15-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑𝑈𝑃)    &   (𝜑𝑉𝑃)    &   (𝜑𝑈 ∈ (𝑋𝐼𝑍))    &   (𝜑𝑉 ∈ (𝑌𝐼𝑍))       (𝜑 → ∃𝑎𝑃 (𝑎 ∈ (𝑈𝐼𝑌) ∧ 𝑎 ∈ (𝑉𝐼𝑋)))
 
Theoremaxtgcont1 26182* Axiom of Continuity. Axiom A11 of [Schwabhauser] p. 13. This axiom (scheme) asserts that any two sets 𝑆 and 𝑇 (of points) such that the elements of 𝑆 precede the elements of 𝑇 with respect to some point 𝑎 (that is, 𝑥 is between 𝑎 and 𝑦 whenever 𝑥 is in 𝑋 and 𝑦 is in 𝑌) are separated by some point 𝑏; this is explained in Axiom 11 of [Tarski1999] p. 185. (Contributed by Thierry Arnoux, 16-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑆𝑃)    &   (𝜑𝑇𝑃)       (𝜑 → (∃𝑎𝑃𝑥𝑆𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑆𝑦𝑇 𝑏 ∈ (𝑥𝐼𝑦)))
 
Theoremaxtgcont 26183* Axiom of Continuity. Axiom A11 of [Schwabhauser] p. 13. For more information see axtgcont1 26182. (Contributed by Thierry Arnoux, 16-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝑆𝑃)    &   (𝜑𝑇𝑃)    &   (𝜑𝐴𝑃)    &   ((𝜑𝑢𝑆𝑣𝑇) → 𝑢 ∈ (𝐴𝐼𝑣))       (𝜑 → ∃𝑏𝑃𝑥𝑆𝑦𝑇 𝑏 ∈ (𝑥𝐼𝑦))
 
Theoremaxtglowdim2 26184* Lower dimension axiom for dimension 2, Axiom A8 of [Schwabhauser] p. 13. There exist 3 non-colinear points. (Contributed by Thierry Arnoux, 20-Nov-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝐺DimTarskiG≥2)       (𝜑 → ∃𝑥𝑃𝑦𝑃𝑧𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))
 
Theoremaxtgupdim2 26185 Upper dimension axiom for dimension 2, Axiom A9 of [Schwabhauser] p. 13. Three points 𝑋, 𝑌 and 𝑍 equidistant to two given two points 𝑈 and 𝑉 must be colinear. (Contributed by Thierry Arnoux, 29-May-2019.) (Revised by Thierry Arnoux, 11-Jul-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑𝑈𝑃)    &   (𝜑𝑉𝑃)    &   (𝜑𝑈𝑉)    &   (𝜑 → (𝑈 𝑋) = (𝑉 𝑋))    &   (𝜑 → (𝑈 𝑌) = (𝑉 𝑌))    &   (𝜑 → (𝑈 𝑍) = (𝑉 𝑍))    &   (𝜑𝐺𝑉)    &   (𝜑 → ¬ 𝐺DimTarskiG≥3)       (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))
 
Theoremaxtgeucl 26186* Euclid's Axiom. Axiom A10 of [Schwabhauser] p. 13. This is equivalent to Euclid's parallel postulate when combined with other axioms. (Contributed by Thierry Arnoux, 16-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiGE)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑𝑈𝑃)    &   (𝜑𝑉𝑃)    &   (𝜑𝑈 ∈ (𝑋𝐼𝑉))    &   (𝜑𝑈 ∈ (𝑌𝐼𝑍))    &   (𝜑𝑋𝑈)       (𝜑 → ∃𝑎𝑃𝑏𝑃 (𝑌 ∈ (𝑋𝐼𝑎) ∧ 𝑍 ∈ (𝑋𝐼𝑏) ∧ 𝑉 ∈ (𝑎𝐼𝑏)))
 
15.1.1  Justification for the congruence notation
 
Theoremtgjustf 26187* Given any function 𝐹, equality of the image by 𝐹 is an equivalence relation. (Contributed by Thierry Arnoux, 25-Jan-2023.)
(𝐴𝑉 → ∃𝑟(𝑟 Er 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑟𝑦 ↔ (𝐹𝑥) = (𝐹𝑦))))
 
Theoremtgjustr 26188* Given any equivalence relation 𝑅, one can define a function 𝑓 such that all elements of an equivalence classe of 𝑅 have the same image by 𝑓. (Contributed by Thierry Arnoux, 25-Jan-2023.)
((𝐴𝑉𝑅 Er 𝐴) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝑓𝑥) = (𝑓𝑦))))
 
Theoremtgjustc1 26189* A justification for using distance equality instead of the textbook relation on pairs of points for congruence. (Contributed by Thierry Arnoux, 29-Jan-2023.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)       𝑟(𝑟 Er (𝑃 × 𝑃) ∧ ∀𝑤𝑃𝑥𝑃𝑦𝑃𝑧𝑃 (⟨𝑤, 𝑥𝑟𝑦, 𝑧⟩ ↔ (𝑤 𝑥) = (𝑦 𝑧)))
 
Theoremtgjustc2 26190* A justification for using distance equality instead of the textbook relation on pairs of points for congruence. (Contributed by Thierry Arnoux, 29-Jan-2023.)
𝑃 = (Base‘𝐺)    &   𝑅 Er (𝑃 × 𝑃)       𝑑(𝑑 Fn (𝑃 × 𝑃) ∧ ∀𝑤𝑃𝑥𝑃𝑦𝑃𝑧𝑃 (⟨𝑤, 𝑥𝑅𝑦, 𝑧⟩ ↔ (𝑤𝑑𝑥) = (𝑦𝑑𝑧)))
 
15.2  Tarskian Geometry
 
15.2.1  Congruence
 
Theoremtgcgrcomimp 26191 Congruence commutes on the RHS. Theorem 2.5 of [Schwabhauser] p. 27. (Contributed by David A. Wheeler, 29-Jun-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)       (𝜑 → ((𝐴 𝐵) = (𝐶 𝐷) → (𝐴 𝐵) = (𝐷 𝐶)))
 
Theoremtgcgrcomr 26192 Congruence commutes on the RHS. Variant of Theorem 2.5 of [Schwabhauser] p. 27, but in a convenient form for a common case. (Contributed by David A. Wheeler, 29-Jun-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))       (𝜑 → (𝐴 𝐵) = (𝐷 𝐶))
 
Theoremtgcgrcoml 26193 Congruence commutes on the LHS. Variant of Theorem 2.5 of [Schwabhauser] p. 27, but in a convenient form for a common case. (Contributed by David A. Wheeler, 29-Jun-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))       (𝜑 → (𝐵 𝐴) = (𝐶 𝐷))
 
Theoremtgcgrcomlr 26194 Congruence commutes on both sides. (Contributed by Thierry Arnoux, 23-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))       (𝜑 → (𝐵 𝐴) = (𝐷 𝐶))
 
Theoremtgcgreqb 26195 Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))       (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))
 
Theoremtgcgreq 26196 Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))    &   (𝜑𝐴 = 𝐵)       (𝜑𝐶 = 𝐷)
 
Theoremtgcgrneq 26197 Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))    &   (𝜑𝐴𝐵)       (𝜑𝐶𝐷)
 
Theoremtgcgrtriv 26198 Degenerate segments are congruent. Theorem 2.8 of [Schwabhauser] p. 28. (Contributed by Thierry Arnoux, 23-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)       (𝜑 → (𝐴 𝐴) = (𝐵 𝐵))
 
Theoremtgcgrextend 26199 Link congruence over a pair of line segments. Theorem 2.11 of [Schwabhauser] p. 29. (Contributed by Thierry Arnoux, 23-Mar-2019.) (Shortened by David A. Wheeler and Thierry Arnoux, 22-Apr-2020.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝐵 ∈ (𝐴𝐼𝐶))    &   (𝜑𝐸 ∈ (𝐷𝐼𝐹))    &   (𝜑 → (𝐴 𝐵) = (𝐷 𝐸))    &   (𝜑 → (𝐵 𝐶) = (𝐸 𝐹))       (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))
 
Theoremtgsegconeq 26200 Two points that satisfy the conclusion of axtgsegcon 26178 are identical. Uniqueness portion of Theorem 2.12 of [Schwabhauser] p. 29. (Contributed by Thierry Arnoux, 23-Mar-2019.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝐷𝐴)    &   (𝜑𝐴 ∈ (𝐷𝐼𝐸))    &   (𝜑𝐴 ∈ (𝐷𝐼𝐹))    &   (𝜑 → (𝐴 𝐸) = (𝐵 𝐶))    &   (𝜑 → (𝐴 𝐹) = (𝐵 𝐶))       (𝜑𝐸 = 𝐹)
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