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Theorem itgsubst 23716
Description: Integration by 𝑢-substitution. If 𝐴(𝑥) is a continuous, differentiable function from [𝑋, 𝑌] to (𝑍, 𝑊), whose derivative is continuous and integrable, and 𝐶(𝑢) is a continuous function on (𝑍, 𝑊), then the integral of 𝐶(𝑢) from 𝐾 = 𝐴(𝑋) to 𝐿 = 𝐴(𝑌) is equal to the integral of 𝐶(𝐴(𝑥)) D 𝐴(𝑥) from 𝑋 to 𝑌. In this part of the proof we discharge the assumptions in itgsubstlem 23715, which use the fact that (𝑍, 𝑊) is open to shrink the interval a little to (𝑀, 𝑁) where 𝑍 < 𝑀 < 𝑁 < 𝑊- this is possible because 𝐴(𝑥) is a continuous function on a closed interval, so its range is in fact a closed interval, and we have some wiggle room on the edges. (Contributed by Mario Carneiro, 7-Sep-2014.)
Hypotheses
Ref Expression
itgsubst.x (𝜑𝑋 ∈ ℝ)
itgsubst.y (𝜑𝑌 ∈ ℝ)
itgsubst.le (𝜑𝑋𝑌)
itgsubst.z (𝜑𝑍 ∈ ℝ*)
itgsubst.w (𝜑𝑊 ∈ ℝ*)
itgsubst.a (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝑍(,)𝑊)))
itgsubst.b (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ (((𝑋(,)𝑌)–cn→ℂ) ∩ 𝐿1))
itgsubst.c (𝜑 → (𝑢 ∈ (𝑍(,)𝑊) ↦ 𝐶) ∈ ((𝑍(,)𝑊)–cn→ℂ))
itgsubst.da (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))
itgsubst.e (𝑢 = 𝐴𝐶 = 𝐸)
itgsubst.k (𝑥 = 𝑋𝐴 = 𝐾)
itgsubst.l (𝑥 = 𝑌𝐴 = 𝐿)
Assertion
Ref Expression
itgsubst (𝜑 → ⨜[𝐾𝐿]𝐶 d𝑢 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥)
Distinct variable groups:   𝑢,𝐸   𝑥,𝑢,𝐾   𝜑,𝑢,𝑥   𝑢,𝑋,𝑥   𝑢,𝑌,𝑥   𝑢,𝐴   𝑥,𝐶   𝑢,𝑊,𝑥   𝑢,𝐿,𝑥   𝑢,𝑍,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥,𝑢)   𝐶(𝑢)   𝐸(𝑥)

Proof of Theorem itgsubst
Dummy variables 𝑚 𝑛 𝑦 𝑧 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 itgsubst.x . . 3 (𝜑𝑋 ∈ ℝ)
2 itgsubst.y . . 3 (𝜑𝑌 ∈ ℝ)
3 itgsubst.le . . 3 (𝜑𝑋𝑌)
4 ioossre 12177 . . . . 5 (𝑍(,)𝑊) ⊆ ℝ
5 ax-resscn 9937 . . . . 5 ℝ ⊆ ℂ
6 cncfss 22610 . . . . 5 (((𝑍(,)𝑊) ⊆ ℝ ∧ ℝ ⊆ ℂ) → ((𝑋[,]𝑌)–cn→(𝑍(,)𝑊)) ⊆ ((𝑋[,]𝑌)–cn→ℝ))
74, 5, 6mp2an 707 . . . 4 ((𝑋[,]𝑌)–cn→(𝑍(,)𝑊)) ⊆ ((𝑋[,]𝑌)–cn→ℝ)
8 itgsubst.a . . . 4 (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝑍(,)𝑊)))
97, 8sseldi 3581 . . 3 (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→ℝ))
101, 2, 3, 9evthicc 23135 . 2 (𝜑 → (∃𝑦 ∈ (𝑋[,]𝑌)∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∧ ∃𝑦 ∈ (𝑋[,]𝑌)∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧)))
11 ressxr 10027 . . . . . . . 8 ℝ ⊆ ℝ*
124, 11sstri 3592 . . . . . . 7 (𝑍(,)𝑊) ⊆ ℝ*
13 cncff 22604 . . . . . . . . . 10 ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝑍(,)𝑊)) → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝑍(,)𝑊))
148, 13syl 17 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝑍(,)𝑊))
1514adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝑍(,)𝑊))
16 simprl 793 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) → 𝑦 ∈ (𝑋[,]𝑌))
1715, 16ffvelrnd 6316 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ (𝑍(,)𝑊))
1812, 17sseldi 3581 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ*)
19 itgsubst.w . . . . . . 7 (𝜑𝑊 ∈ ℝ*)
2019adantr 481 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) → 𝑊 ∈ ℝ*)
21 eliooord 12175 . . . . . . . 8 (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ (𝑍(,)𝑊) → (𝑍 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∧ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑊))
2217, 21syl 17 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) → (𝑍 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∧ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑊))
2322simprd 479 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑊)
24 qbtwnxr 11974 . . . . . 6 ((((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ*𝑊 ∈ ℝ* ∧ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑊) → ∃𝑛 ∈ ℚ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))
2518, 20, 23, 24syl3anc 1323 . . . . 5 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) → ∃𝑛 ∈ ℚ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))
26 qre 11737 . . . . . . . . . 10 (𝑛 ∈ ℚ → 𝑛 ∈ ℝ)
2726ad2antrl 763 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → 𝑛 ∈ ℝ)
28 itgsubst.z . . . . . . . . . . 11 (𝜑𝑍 ∈ ℝ*)
2928ad2antrr 761 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → 𝑍 ∈ ℝ*)
3018adantr 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ*)
3127rexrd 10033 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → 𝑛 ∈ ℝ*)
3222simpld 475 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) → 𝑍 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))
3332adantr 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → 𝑍 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))
34 simprrl 803 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛)
3529, 30, 31, 33, 34xrlttrd 11934 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → 𝑍 < 𝑛)
36 simprrr 804 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → 𝑛 < 𝑊)
3719ad2antrr 761 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → 𝑊 ∈ ℝ*)
38 elioo2 12158 . . . . . . . . . 10 ((𝑍 ∈ ℝ*𝑊 ∈ ℝ*) → (𝑛 ∈ (𝑍(,)𝑊) ↔ (𝑛 ∈ ℝ ∧ 𝑍 < 𝑛𝑛 < 𝑊)))
3929, 37, 38syl2anc 692 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → (𝑛 ∈ (𝑍(,)𝑊) ↔ (𝑛 ∈ ℝ ∧ 𝑍 < 𝑛𝑛 < 𝑊)))
4027, 35, 36, 39mpbir3and 1243 . . . . . . . 8 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → 𝑛 ∈ (𝑍(,)𝑊))
41 anass 680 . . . . . . . . 9 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)) ↔ (𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))))
42 simprrl 803 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛)
4342adantr 481 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛)
4414ad2antrr 761 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝑍(,)𝑊))
4544ffvelrnda 6315 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑍(,)𝑊))
4612, 45sseldi 3581 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ ℝ*)
47 simplr 791 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → 𝑦 ∈ (𝑋[,]𝑌))
4844, 47ffvelrnd 6316 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ (𝑍(,)𝑊))
4912, 48sseldi 3581 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ*)
5049adantr 481 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ*)
5126ad2antrl 763 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → 𝑛 ∈ ℝ)
5251adantr 481 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑛 ∈ ℝ)
5352rexrd 10033 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑛 ∈ ℝ*)
54 xrlelttr 11931 . . . . . . . . . . . . . 14 ((((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ ℝ* ∧ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ*𝑛 ∈ ℝ*) → ((((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∧ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛))
5546, 50, 53, 54syl3anc 1323 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∧ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛))
5643, 55mpan2d 709 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛))
5756ralimdva 2956 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → (∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) → ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛))
5857imp 445 . . . . . . . . . 10 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)) → ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛)
5958an32s 845 . . . . . . . . 9 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛)
6041, 59sylanbr 490 . . . . . . . 8 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛)
6140, 60jca 554 . . . . . . 7 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → (𝑛 ∈ (𝑍(,)𝑊) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛))
6261ex 450 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) → ((𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊)) → (𝑛 ∈ (𝑍(,)𝑊) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛)))
6362reximdv2 3008 . . . . 5 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) → (∃𝑛 ∈ ℚ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊) → ∃𝑛 ∈ (𝑍(,)𝑊)∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛))
6425, 63mpd 15 . . . 4 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) → ∃𝑛 ∈ (𝑍(,)𝑊)∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛)
6564rexlimdvaa 3025 . . 3 (𝜑 → (∃𝑦 ∈ (𝑋[,]𝑌)∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) → ∃𝑛 ∈ (𝑍(,)𝑊)∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛))
6628adantr 481 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) → 𝑍 ∈ ℝ*)
6714adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝑍(,)𝑊))
68 simprl 793 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) → 𝑦 ∈ (𝑋[,]𝑌))
6967, 68ffvelrnd 6316 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ (𝑍(,)𝑊))
7012, 69sseldi 3581 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ*)
7169, 21syl 17 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) → (𝑍 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∧ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑊))
7271simpld 475 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) → 𝑍 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))
73 qbtwnxr 11974 . . . . . 6 ((𝑍 ∈ ℝ* ∧ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ*𝑍 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)) → ∃𝑚 ∈ ℚ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))
7466, 70, 72, 73syl3anc 1323 . . . . 5 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) → ∃𝑚 ∈ ℚ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))
75 qre 11737 . . . . . . . . . 10 (𝑚 ∈ ℚ → 𝑚 ∈ ℝ)
7675ad2antrl 763 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑚 ∈ ℝ)
77 simprrl 803 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑍 < 𝑚)
7876rexrd 10033 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑚 ∈ ℝ*)
7970adantr 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ*)
8019ad2antrr 761 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑊 ∈ ℝ*)
81 simprrr 804 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))
8271simprd 479 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑊)
8382adantr 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑊)
8478, 79, 80, 81, 83xrlttrd 11934 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑚 < 𝑊)
8528ad2antrr 761 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑍 ∈ ℝ*)
86 elioo2 12158 . . . . . . . . . 10 ((𝑍 ∈ ℝ*𝑊 ∈ ℝ*) → (𝑚 ∈ (𝑍(,)𝑊) ↔ (𝑚 ∈ ℝ ∧ 𝑍 < 𝑚𝑚 < 𝑊)))
8785, 80, 86syl2anc 692 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → (𝑚 ∈ (𝑍(,)𝑊) ↔ (𝑚 ∈ ℝ ∧ 𝑍 < 𝑚𝑚 < 𝑊)))
8876, 77, 84, 87mpbir3and 1243 . . . . . . . 8 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑚 ∈ (𝑍(,)𝑊))
89 anass 680 . . . . . . . . 9 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧)) ↔ (𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))))
90 simprrr 804 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))
9190adantr 481 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))
9275ad2antrl 763 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑚 ∈ ℝ)
9392adantr 481 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑚 ∈ ℝ)
9493rexrd 10033 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑚 ∈ ℝ*)
9514ad2antrr 761 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝑍(,)𝑊))
96 simplr 791 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑦 ∈ (𝑋[,]𝑌))
9795, 96ffvelrnd 6316 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ (𝑍(,)𝑊))
9812, 97sseldi 3581 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ*)
9998adantr 481 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ*)
10095ffvelrnda 6315 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑍(,)𝑊))
10112, 100sseldi 3581 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ ℝ*)
102 xrltletr 11932 . . . . . . . . . . . . . 14 ((𝑚 ∈ ℝ* ∧ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ* ∧ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ ℝ*) → ((𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∧ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧)) → 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧)))
10394, 99, 101, 102syl3anc 1323 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∧ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧)) → 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧)))
10491, 103mpand 710 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) → 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧)))
105104ralimdva 2956 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → (∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) → ∀𝑧 ∈ (𝑋[,]𝑌)𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧)))
106105imp 445 . . . . . . . . . 10 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧)) → ∀𝑧 ∈ (𝑋[,]𝑌)𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))
107106an32s 845 . . . . . . . . 9 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → ∀𝑧 ∈ (𝑋[,]𝑌)𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))
10889, 107sylanbr 490 . . . . . . . 8 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → ∀𝑧 ∈ (𝑋[,]𝑌)𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))
10988, 108jca 554 . . . . . . 7 (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → (𝑚 ∈ (𝑍(,)𝑊) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧)))
110109ex 450 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) → ((𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) → (𝑚 ∈ (𝑍(,)𝑊) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))))
111110reximdv2 3008 . . . . 5 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) → (∃𝑚 ∈ ℚ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)) → ∃𝑚 ∈ (𝑍(,)𝑊)∀𝑧 ∈ (𝑋[,]𝑌)𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧)))
11274, 111mpd 15 . . . 4 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) → ∃𝑚 ∈ (𝑍(,)𝑊)∀𝑧 ∈ (𝑋[,]𝑌)𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))
113112rexlimdvaa 3025 . . 3 (𝜑 → (∃𝑦 ∈ (𝑋[,]𝑌)∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) → ∃𝑚 ∈ (𝑍(,)𝑊)∀𝑧 ∈ (𝑋[,]𝑌)𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧)))
114 ancom 466 . . . . 5 ((∃𝑛 ∈ (𝑍(,)𝑊)∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛 ∧ ∃𝑚 ∈ (𝑍(,)𝑊)∀𝑧 ∈ (𝑋[,]𝑌)𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧)) ↔ (∃𝑚 ∈ (𝑍(,)𝑊)∀𝑧 ∈ (𝑋[,]𝑌)𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∧ ∃𝑛 ∈ (𝑍(,)𝑊)∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛))
115 reeanv 3097 . . . . 5 (∃𝑚 ∈ (𝑍(,)𝑊)∃𝑛 ∈ (𝑍(,)𝑊)(∀𝑧 ∈ (𝑋[,]𝑌)𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛) ↔ (∃𝑚 ∈ (𝑍(,)𝑊)∀𝑧 ∈ (𝑋[,]𝑌)𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∧ ∃𝑛 ∈ (𝑍(,)𝑊)∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛))
116114, 115bitr4i 267 . . . 4 ((∃𝑛 ∈ (𝑍(,)𝑊)∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛 ∧ ∃𝑚 ∈ (𝑍(,)𝑊)∀𝑧 ∈ (𝑋[,]𝑌)𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧)) ↔ ∃𝑚 ∈ (𝑍(,)𝑊)∃𝑛 ∈ (𝑍(,)𝑊)(∀𝑧 ∈ (𝑋[,]𝑌)𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛))
117 r19.26 3057 . . . . . 6 (∀𝑧 ∈ (𝑋[,]𝑌)(𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∧ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛) ↔ (∀𝑧 ∈ (𝑋[,]𝑌)𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛))
11814adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝑍(,)𝑊))
119118ffvelrnda 6315 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑍(,)𝑊))
1204, 119sseldi 3581 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ ℝ)
1211203biant1d 1438 . . . . . . . . 9 (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∧ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛) ↔ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ ℝ ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∧ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛)))
122 simplrl 799 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑚 ∈ (𝑍(,)𝑊))
12312, 122sseldi 3581 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑚 ∈ ℝ*)
124 simplrr 800 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑛 ∈ (𝑍(,)𝑊))
12512, 124sseldi 3581 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑛 ∈ ℝ*)
126 elioo2 12158 . . . . . . . . . 10 ((𝑚 ∈ ℝ*𝑛 ∈ ℝ*) → (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑚(,)𝑛) ↔ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ ℝ ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∧ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛)))
127123, 125, 126syl2anc 692 . . . . . . . . 9 (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑚(,)𝑛) ↔ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ ℝ ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∧ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛)))
128121, 127bitr4d 271 . . . . . . . 8 (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∧ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛) ↔ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑚(,)𝑛)))
129128ralbidva 2979 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) → (∀𝑧 ∈ (𝑋[,]𝑌)(𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∧ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛) ↔ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑚(,)𝑛)))
130 nffvmpt1 6156 . . . . . . . . . . . 12 𝑥((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧)
131130nfel1 2775 . . . . . . . . . . 11 𝑥((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑚(,)𝑛)
132 nfv 1840 . . . . . . . . . . 11 𝑧((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) ∈ (𝑚(,)𝑛)
133 fveq2 6148 . . . . . . . . . . . 12 (𝑧 = 𝑥 → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) = ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥))
134133eleq1d 2683 . . . . . . . . . . 11 (𝑧 = 𝑥 → (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑚(,)𝑛) ↔ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) ∈ (𝑚(,)𝑛)))
135131, 132, 134cbvral 3155 . . . . . . . . . 10 (∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑚(,)𝑛) ↔ ∀𝑥 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) ∈ (𝑚(,)𝑛))
136 simpr 477 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝑋[,]𝑌)) → 𝑥 ∈ (𝑋[,]𝑌))
137 eqid 2621 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) = (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)
138137fmpt 6337 . . . . . . . . . . . . . . 15 (∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑍(,)𝑊) ↔ (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝑍(,)𝑊))
13914, 138sylibr 224 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑍(,)𝑊))
140139r19.21bi 2927 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝑋[,]𝑌)) → 𝐴 ∈ (𝑍(,)𝑊))
141137fvmpt2 6248 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑋[,]𝑌) ∧ 𝐴 ∈ (𝑍(,)𝑊)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) = 𝐴)
142136, 140, 141syl2anc 692 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) = 𝐴)
143142eleq1d 2683 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝑋[,]𝑌)) → (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) ∈ (𝑚(,)𝑛) ↔ 𝐴 ∈ (𝑚(,)𝑛)))
144143ralbidva 2979 . . . . . . . . . 10 (𝜑 → (∀𝑥 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) ∈ (𝑚(,)𝑛) ↔ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛)))
145135, 144syl5bb 272 . . . . . . . . 9 (𝜑 → (∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑚(,)𝑛) ↔ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛)))
146145adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) → (∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑚(,)𝑛) ↔ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛)))
1471adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑋 ∈ ℝ)
1482adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑌 ∈ ℝ)
1493adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑋𝑌)
15028adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑍 ∈ ℝ*)
15119adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑊 ∈ ℝ*)
152 nfcv 2761 . . . . . . . . . . . . . 14 𝑦𝐴
153 nfcsb1v 3530 . . . . . . . . . . . . . 14 𝑥𝑦 / 𝑥𝐴
154 csbeq1a 3523 . . . . . . . . . . . . . 14 (𝑥 = 𝑦𝐴 = 𝑦 / 𝑥𝐴)
155152, 153, 154cbvmpt 4709 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) = (𝑦 ∈ (𝑋[,]𝑌) ↦ 𝑦 / 𝑥𝐴)
156155, 8syl5eqelr 2703 . . . . . . . . . . . 12 (𝜑 → (𝑦 ∈ (𝑋[,]𝑌) ↦ 𝑦 / 𝑥𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝑍(,)𝑊)))
157156adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → (𝑦 ∈ (𝑋[,]𝑌) ↦ 𝑦 / 𝑥𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝑍(,)𝑊)))
158 nfcv 2761 . . . . . . . . . . . . . 14 𝑦𝐵
159 nfcsb1v 3530 . . . . . . . . . . . . . 14 𝑥𝑦 / 𝑥𝐵
160 csbeq1a 3523 . . . . . . . . . . . . . 14 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
161158, 159, 160cbvmpt 4709 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) = (𝑦 ∈ (𝑋(,)𝑌) ↦ 𝑦 / 𝑥𝐵)
162 itgsubst.b . . . . . . . . . . . . 13 (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ (((𝑋(,)𝑌)–cn→ℂ) ∩ 𝐿1))
163161, 162syl5eqelr 2703 . . . . . . . . . . . 12 (𝜑 → (𝑦 ∈ (𝑋(,)𝑌) ↦ 𝑦 / 𝑥𝐵) ∈ (((𝑋(,)𝑌)–cn→ℂ) ∩ 𝐿1))
164163adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → (𝑦 ∈ (𝑋(,)𝑌) ↦ 𝑦 / 𝑥𝐵) ∈ (((𝑋(,)𝑌)–cn→ℂ) ∩ 𝐿1))
165 nfcv 2761 . . . . . . . . . . . . . 14 𝑣𝐶
166 nfcsb1v 3530 . . . . . . . . . . . . . 14 𝑢𝑣 / 𝑢𝐶
167 csbeq1a 3523 . . . . . . . . . . . . . 14 (𝑢 = 𝑣𝐶 = 𝑣 / 𝑢𝐶)
168165, 166, 167cbvmpt 4709 . . . . . . . . . . . . 13 (𝑢 ∈ (𝑍(,)𝑊) ↦ 𝐶) = (𝑣 ∈ (𝑍(,)𝑊) ↦ 𝑣 / 𝑢𝐶)
169 itgsubst.c . . . . . . . . . . . . 13 (𝜑 → (𝑢 ∈ (𝑍(,)𝑊) ↦ 𝐶) ∈ ((𝑍(,)𝑊)–cn→ℂ))
170168, 169syl5eqelr 2703 . . . . . . . . . . . 12 (𝜑 → (𝑣 ∈ (𝑍(,)𝑊) ↦ 𝑣 / 𝑢𝐶) ∈ ((𝑍(,)𝑊)–cn→ℂ))
171170adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → (𝑣 ∈ (𝑍(,)𝑊) ↦ 𝑣 / 𝑢𝐶) ∈ ((𝑍(,)𝑊)–cn→ℂ))
172 itgsubst.da . . . . . . . . . . . . 13 (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))
173155oveq2i 6615 . . . . . . . . . . . . 13 (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (ℝ D (𝑦 ∈ (𝑋[,]𝑌) ↦ 𝑦 / 𝑥𝐴))
174172, 173, 1613eqtr3g 2678 . . . . . . . . . . . 12 (𝜑 → (ℝ D (𝑦 ∈ (𝑋[,]𝑌) ↦ 𝑦 / 𝑥𝐴)) = (𝑦 ∈ (𝑋(,)𝑌) ↦ 𝑦 / 𝑥𝐵))
175174adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → (ℝ D (𝑦 ∈ (𝑋[,]𝑌) ↦ 𝑦 / 𝑥𝐴)) = (𝑦 ∈ (𝑋(,)𝑌) ↦ 𝑦 / 𝑥𝐵))
176 csbeq1 3517 . . . . . . . . . . 11 (𝑣 = 𝑦 / 𝑥𝐴𝑣 / 𝑢𝐶 = 𝑦 / 𝑥𝐴 / 𝑢𝐶)
177 csbeq1 3517 . . . . . . . . . . 11 (𝑦 = 𝑋𝑦 / 𝑥𝐴 = 𝑋 / 𝑥𝐴)
178 csbeq1 3517 . . . . . . . . . . 11 (𝑦 = 𝑌𝑦 / 𝑥𝐴 = 𝑌 / 𝑥𝐴)
179 simprll 801 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑚 ∈ (𝑍(,)𝑊))
180 simprlr 802 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑛 ∈ (𝑍(,)𝑊))
181 simprr 795 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))
182153nfel1 2775 . . . . . . . . . . . . 13 𝑥𝑦 / 𝑥𝐴 ∈ (𝑚(,)𝑛)
183154eleq1d 2683 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝐴 ∈ (𝑚(,)𝑛) ↔ 𝑦 / 𝑥𝐴 ∈ (𝑚(,)𝑛)))
184182, 183rspc 3289 . . . . . . . . . . . 12 (𝑦 ∈ (𝑋[,]𝑌) → (∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛) → 𝑦 / 𝑥𝐴 ∈ (𝑚(,)𝑛)))
185181, 184mpan9 486 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) ∧ 𝑦 ∈ (𝑋[,]𝑌)) → 𝑦 / 𝑥𝐴 ∈ (𝑚(,)𝑛))
186147, 148, 149, 150, 151, 157, 164, 171, 175, 176, 177, 178, 179, 180, 185itgsubstlem 23715 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → ⨜[𝑋 / 𝑥𝐴𝑌 / 𝑥𝐴]𝑣 / 𝑢𝐶 d𝑣 = ⨜[𝑋𝑌](𝑦 / 𝑥𝐴 / 𝑢𝐶 · 𝑦 / 𝑥𝐵) d𝑦)
187167, 165, 166cbvditg 23524 . . . . . . . . . . . 12 ⨜[𝑋 / 𝑥𝐴𝑌 / 𝑥𝐴]𝐶 d𝑢 = ⨜[𝑋 / 𝑥𝐴𝑌 / 𝑥𝐴]𝑣 / 𝑢𝐶 d𝑣
188 nfcvd 2762 . . . . . . . . . . . . . . 15 (𝑋 ∈ ℝ → 𝑥𝐾)
189 itgsubst.k . . . . . . . . . . . . . . 15 (𝑥 = 𝑋𝐴 = 𝐾)
190188, 189csbiegf 3538 . . . . . . . . . . . . . 14 (𝑋 ∈ ℝ → 𝑋 / 𝑥𝐴 = 𝐾)
191 ditgeq1 23518 . . . . . . . . . . . . . 14 (𝑋 / 𝑥𝐴 = 𝐾 → ⨜[𝑋 / 𝑥𝐴𝑌 / 𝑥𝐴]𝐶 d𝑢 = ⨜[𝐾𝑌 / 𝑥𝐴]𝐶 d𝑢)
1921, 190, 1913syl 18 . . . . . . . . . . . . 13 (𝜑 → ⨜[𝑋 / 𝑥𝐴𝑌 / 𝑥𝐴]𝐶 d𝑢 = ⨜[𝐾𝑌 / 𝑥𝐴]𝐶 d𝑢)
193 nfcvd 2762 . . . . . . . . . . . . . . 15 (𝑌 ∈ ℝ → 𝑥𝐿)
194 itgsubst.l . . . . . . . . . . . . . . 15 (𝑥 = 𝑌𝐴 = 𝐿)
195193, 194csbiegf 3538 . . . . . . . . . . . . . 14 (𝑌 ∈ ℝ → 𝑌 / 𝑥𝐴 = 𝐿)
196 ditgeq2 23519 . . . . . . . . . . . . . 14 (𝑌 / 𝑥𝐴 = 𝐿 → ⨜[𝐾𝑌 / 𝑥𝐴]𝐶 d𝑢 = ⨜[𝐾𝐿]𝐶 d𝑢)
1972, 195, 1963syl 18 . . . . . . . . . . . . 13 (𝜑 → ⨜[𝐾𝑌 / 𝑥𝐴]𝐶 d𝑢 = ⨜[𝐾𝐿]𝐶 d𝑢)
198192, 197eqtrd 2655 . . . . . . . . . . . 12 (𝜑 → ⨜[𝑋 / 𝑥𝐴𝑌 / 𝑥𝐴]𝐶 d𝑢 = ⨜[𝐾𝐿]𝐶 d𝑢)
199187, 198syl5eqr 2669 . . . . . . . . . . 11 (𝜑 → ⨜[𝑋 / 𝑥𝐴𝑌 / 𝑥𝐴]𝑣 / 𝑢𝐶 d𝑣 = ⨜[𝐾𝐿]𝐶 d𝑢)
200199adantr 481 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → ⨜[𝑋 / 𝑥𝐴𝑌 / 𝑥𝐴]𝑣 / 𝑢𝐶 d𝑣 = ⨜[𝐾𝐿]𝐶 d𝑢)
201154csbeq1d 3521 . . . . . . . . . . . . . 14 (𝑥 = 𝑦𝐴 / 𝑢𝐶 = 𝑦 / 𝑥𝐴 / 𝑢𝐶)
202201, 160oveq12d 6622 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝐴 / 𝑢𝐶 · 𝐵) = (𝑦 / 𝑥𝐴 / 𝑢𝐶 · 𝑦 / 𝑥𝐵))
203 nfcv 2761 . . . . . . . . . . . . 13 𝑦(𝐴 / 𝑢𝐶 · 𝐵)
204 nfcv 2761 . . . . . . . . . . . . . . 15 𝑥𝐶
205153, 204nfcsb 3532 . . . . . . . . . . . . . 14 𝑥𝑦 / 𝑥𝐴 / 𝑢𝐶
206 nfcv 2761 . . . . . . . . . . . . . 14 𝑥 ·
207205, 206, 159nfov 6630 . . . . . . . . . . . . 13 𝑥(𝑦 / 𝑥𝐴 / 𝑢𝐶 · 𝑦 / 𝑥𝐵)
208202, 203, 207cbvditg 23524 . . . . . . . . . . . 12 ⨜[𝑋𝑌](𝐴 / 𝑢𝐶 · 𝐵) d𝑥 = ⨜[𝑋𝑌](𝑦 / 𝑥𝐴 / 𝑢𝐶 · 𝑦 / 𝑥𝐵) d𝑦
209 ioossicc 12201 . . . . . . . . . . . . . . . . . 18 (𝑋(,)𝑌) ⊆ (𝑋[,]𝑌)
210209sseli 3579 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝑋(,)𝑌) → 𝑥 ∈ (𝑋[,]𝑌))
211210, 140sylan2 491 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ (𝑍(,)𝑊))
212 nfcvd 2762 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ (𝑍(,)𝑊) → 𝑢𝐸)
213 itgsubst.e . . . . . . . . . . . . . . . . 17 (𝑢 = 𝐴𝐶 = 𝐸)
214212, 213csbiegf 3538 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (𝑍(,)𝑊) → 𝐴 / 𝑢𝐶 = 𝐸)
215211, 214syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 / 𝑢𝐶 = 𝐸)
216215oveq1d 6619 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → (𝐴 / 𝑢𝐶 · 𝐵) = (𝐸 · 𝐵))
217216itgeq2dv 23454 . . . . . . . . . . . . 13 (𝜑 → ∫(𝑋(,)𝑌)(𝐴 / 𝑢𝐶 · 𝐵) d𝑥 = ∫(𝑋(,)𝑌)(𝐸 · 𝐵) d𝑥)
2183ditgpos 23526 . . . . . . . . . . . . 13 (𝜑 → ⨜[𝑋𝑌](𝐴 / 𝑢𝐶 · 𝐵) d𝑥 = ∫(𝑋(,)𝑌)(𝐴 / 𝑢𝐶 · 𝐵) d𝑥)
2193ditgpos 23526 . . . . . . . . . . . . 13 (𝜑 → ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥 = ∫(𝑋(,)𝑌)(𝐸 · 𝐵) d𝑥)
220217, 218, 2193eqtr4d 2665 . . . . . . . . . . . 12 (𝜑 → ⨜[𝑋𝑌](𝐴 / 𝑢𝐶 · 𝐵) d𝑥 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥)
221208, 220syl5eqr 2669 . . . . . . . . . . 11 (𝜑 → ⨜[𝑋𝑌](𝑦 / 𝑥𝐴 / 𝑢𝐶 · 𝑦 / 𝑥𝐵) d𝑦 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥)
222221adantr 481 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → ⨜[𝑋𝑌](𝑦 / 𝑥𝐴 / 𝑢𝐶 · 𝑦 / 𝑥𝐵) d𝑦 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥)
223186, 200, 2223eqtr3d 2663 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → ⨜[𝐾𝐿]𝐶 d𝑢 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥)
224223expr 642 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) → (∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛) → ⨜[𝐾𝐿]𝐶 d𝑢 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥))
225146, 224sylbid 230 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) → (∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑚(,)𝑛) → ⨜[𝐾𝐿]𝐶 d𝑢 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥))
226129, 225sylbid 230 . . . . . 6 ((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) → (∀𝑧 ∈ (𝑋[,]𝑌)(𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∧ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛) → ⨜[𝐾𝐿]𝐶 d𝑢 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥))
227117, 226syl5bir 233 . . . . 5 ((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) → ((∀𝑧 ∈ (𝑋[,]𝑌)𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛) → ⨜[𝐾𝐿]𝐶 d𝑢 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥))
228227rexlimdvva 3031 . . . 4 (𝜑 → (∃𝑚 ∈ (𝑍(,)𝑊)∃𝑛 ∈ (𝑍(,)𝑊)(∀𝑧 ∈ (𝑋[,]𝑌)𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛) → ⨜[𝐾𝐿]𝐶 d𝑢 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥))
229116, 228syl5bi 232 . . 3 (𝜑 → ((∃𝑛 ∈ (𝑍(,)𝑊)∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛 ∧ ∃𝑚 ∈ (𝑍(,)𝑊)∀𝑧 ∈ (𝑋[,]𝑌)𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧)) → ⨜[𝐾𝐿]𝐶 d𝑢 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥))
23065, 113, 229syl2and 500 . 2 (𝜑 → ((∃𝑦 ∈ (𝑋[,]𝑌)∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∧ ∃𝑦 ∈ (𝑋[,]𝑌)∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧)) → ⨜[𝐾𝐿]𝐶 d𝑢 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥))
23110, 230mpd 15 1 (𝜑 → ⨜[𝐾𝐿]𝐶 d𝑢 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  csb 3514  cin 3554  wss 3555   class class class wbr 4613  cmpt 4673  wf 5843  cfv 5847  (class class class)co 6604  cc 9878  cr 9879   · cmul 9885  *cxr 10017   < clt 10018  cle 10019  cq 11732  (,)cioo 12117  [,]cicc 12120  cnccncf 22587  𝐿1cibl 23292  citg 23293  cdit 23516   D cdv 23533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cc 9201  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958  ax-addf 9959  ax-mulf 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-disj 4584  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-ofr 6851  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-omul 7510  df-er 7687  df-map 7804  df-pm 7805  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-fi 8261  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-acn 8712  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ioo 12121  df-ioc 12122  df-ico 12123  df-icc 12124  df-fz 12269  df-fzo 12407  df-fl 12533  df-mod 12609  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-limsup 14136  df-clim 14153  df-rlim 14154  df-sum 14351  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-starv 15877  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-unif 15886  df-hom 15887  df-cco 15888  df-rest 16004  df-topn 16005  df-0g 16023  df-gsum 16024  df-topgen 16025  df-pt 16026  df-prds 16029  df-xrs 16083  df-qtop 16088  df-imas 16089  df-xps 16091  df-mre 16167  df-mrc 16168  df-acs 16170  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-mulg 17462  df-cntz 17671  df-cmn 18116  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-fbas 19662  df-fg 19663  df-cnfld 19666  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-cld 20733  df-ntr 20734  df-cls 20735  df-nei 20812  df-lp 20850  df-perf 20851  df-cn 20941  df-cnp 20942  df-haus 21029  df-cmp 21100  df-tx 21275  df-hmeo 21468  df-fil 21560  df-fm 21652  df-flim 21653  df-flf 21654  df-xms 22035  df-ms 22036  df-tms 22037  df-cncf 22589  df-ovol 23140  df-vol 23141  df-mbf 23294  df-itg1 23295  df-itg2 23296  df-ibl 23297  df-itg 23298  df-0p 23343  df-ditg 23517  df-limc 23536  df-dv 23537
This theorem is referenced by:  itgsubsticclem  39498
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