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Theorem axpre-mulext 7019
 Description: Strong extensionality of multiplication (expressed in terms of <ℝ). Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulext 7059. (Contributed by Jim Kingdon, 18-Feb-2020.) (New usage is discouraged.)
Assertion
Ref Expression
axpre-mulext ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐶) < (𝐵 · 𝐶) → (𝐴 < 𝐵𝐵 < 𝐴)))

Proof of Theorem axpre-mulext
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elreal 6962 . 2 (𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
2 elreal 6962 . 2 (𝐵 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐵)
3 elreal 6962 . 2 (𝐶 ∈ ℝ ↔ ∃𝑧R𝑧, 0R⟩ = 𝐶)
4 oveq1 5546 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) = (𝐴 · ⟨𝑧, 0R⟩))
54breq1d 3801 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) ↔ (𝐴 · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩)))
6 breq1 3794 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝐴 <𝑦, 0R⟩))
7 breq2 3795 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ <𝑥, 0R⟩ ↔ ⟨𝑦, 0R⟩ < 𝐴))
86, 7orbi12d 717 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ (𝐴 <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴)))
95, 8imbi12d 227 . 2 (⟨𝑥, 0R⟩ = 𝐴 → (((⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩)) ↔ ((𝐴 · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) → (𝐴 <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴))))
10 oveq1 5546 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = (𝐵 · ⟨𝑧, 0R⟩))
1110breq2d 3803 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) ↔ (𝐴 · ⟨𝑧, 0R⟩) < (𝐵 · ⟨𝑧, 0R⟩)))
12 breq2 3795 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <𝑦, 0R⟩ ↔ 𝐴 < 𝐵))
13 breq1 3794 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ < 𝐴𝐵 < 𝐴))
1412, 13orbi12d 717 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴) ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
1511, 14imbi12d 227 . 2 (⟨𝑦, 0R⟩ = 𝐵 → (((𝐴 · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) → (𝐴 <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴)) ↔ ((𝐴 · ⟨𝑧, 0R⟩) < (𝐵 · ⟨𝑧, 0R⟩) → (𝐴 < 𝐵𝐵 < 𝐴))))
16 oveq2 5547 . . . 4 (⟨𝑧, 0R⟩ = 𝐶 → (𝐴 · ⟨𝑧, 0R⟩) = (𝐴 · 𝐶))
17 oveq2 5547 . . . 4 (⟨𝑧, 0R⟩ = 𝐶 → (𝐵 · ⟨𝑧, 0R⟩) = (𝐵 · 𝐶))
1816, 17breq12d 3804 . . 3 (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 · ⟨𝑧, 0R⟩) < (𝐵 · ⟨𝑧, 0R⟩) ↔ (𝐴 · 𝐶) < (𝐵 · 𝐶)))
1918imbi1d 224 . 2 (⟨𝑧, 0R⟩ = 𝐶 → (((𝐴 · ⟨𝑧, 0R⟩) < (𝐵 · ⟨𝑧, 0R⟩) → (𝐴 < 𝐵𝐵 < 𝐴)) ↔ ((𝐴 · 𝐶) < (𝐵 · 𝐶) → (𝐴 < 𝐵𝐵 < 𝐴))))
20 mulextsr1 6922 . . 3 ((𝑥R𝑦R𝑧R) → ((𝑥 ·R 𝑧) <R (𝑦 ·R 𝑧) → (𝑥 <R 𝑦𝑦 <R 𝑥)))
21 mulresr 6971 . . . . . 6 ((𝑥R𝑧R) → (⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑥 ·R 𝑧), 0R⟩)
22213adant2 934 . . . . 5 ((𝑥R𝑦R𝑧R) → (⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑥 ·R 𝑧), 0R⟩)
23 mulresr 6971 . . . . . 6 ((𝑦R𝑧R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
24233adant1 933 . . . . 5 ((𝑥R𝑦R𝑧R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
2522, 24breq12d 3804 . . . 4 ((𝑥R𝑦R𝑧R) → ((⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) ↔ ⟨(𝑥 ·R 𝑧), 0R⟩ < ⟨(𝑦 ·R 𝑧), 0R⟩))
26 ltresr 6972 . . . 4 (⟨(𝑥 ·R 𝑧), 0R⟩ < ⟨(𝑦 ·R 𝑧), 0R⟩ ↔ (𝑥 ·R 𝑧) <R (𝑦 ·R 𝑧))
2725, 26syl6bb 189 . . 3 ((𝑥R𝑦R𝑧R) → ((⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) ↔ (𝑥 ·R 𝑧) <R (𝑦 ·R 𝑧)))
28 ltresr 6972 . . . . 5 (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
29 ltresr 6972 . . . . 5 (⟨𝑦, 0R⟩ <𝑥, 0R⟩ ↔ 𝑦 <R 𝑥)
3028, 29orbi12i 691 . . . 4 ((⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ (𝑥 <R 𝑦𝑦 <R 𝑥))
3130a1i 9 . . 3 ((𝑥R𝑦R𝑧R) → ((⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ (𝑥 <R 𝑦𝑦 <R 𝑥)))
3220, 27, 313imtr4d 196 . 2 ((𝑥R𝑦R𝑧R) → ((⟨𝑥, 0R⟩ · ⟨𝑧, 0R⟩) < (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩)))
331, 2, 3, 9, 15, 19, 323gencl 2605 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐶) < (𝐵 · 𝐶) → (𝐴 < 𝐵𝐵 < 𝐴)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 102   ∨ wo 639   ∧ w3a 896   = wceq 1259   ∈ wcel 1409  ⟨cop 3405   class class class wbr 3791  (class class class)co 5539  Rcnr 6452  0Rc0r 6453   ·R cmr 6457
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