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Theorem itgsubst 23616
 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 23615, 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 12106 . . . . 5 (𝑍(,)𝑊) ⊆ ℝ
5 ax-resscn 9872 . . . . 5 ℝ ⊆ ℂ
6 cncfss 22510 . . . . 5 (((𝑍(,)𝑊) ⊆ ℝ ∧ ℝ ⊆ ℂ) → ((𝑋[,]𝑌)–cn→(𝑍(,)𝑊)) ⊆ ((𝑋[,]𝑌)–cn→ℝ))
74, 5, 6mp2an 704 . . . 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 3566 . . 3 (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→ℝ))
101, 2, 3, 9evthicc 23035 . 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 9962 . . . . . . . 8 ℝ ⊆ ℝ*
124, 11sstri 3577 . . . . . . 7 (𝑍(,)𝑊) ⊆ ℝ*
13 cncff 22504 . . . . . . . . . 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 480 . . . . . . . 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 790 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) → 𝑦 ∈ (𝑋[,]𝑌))
1715, 16ffvelrnd 6268 . . . . . . 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 3566 . . . . . 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 480 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) → 𝑊 ∈ ℝ*)
21 eliooord 12104 . . . . . . . 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 478 . . . . . 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 11905 . . . . . 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 1318 . . . . 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 11669 . . . . . . . . . 10 (𝑛 ∈ ℚ → 𝑛 ∈ ℝ)
2726ad2antrl 760 . . . . . . . . 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 758 . . . . . . . . . 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 480 . . . . . . . . . 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 9968 . . . . . . . . . 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 474 . . . . . . . . . . 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 480 . . . . . . . . . 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 800 . . . . . . . . . 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 11866 . . . . . . . . 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 801 . . . . . . . . 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 758 . . . . . . . . . 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 12087 . . . . . . . . . 10 ((𝑍 ∈ ℝ*𝑊 ∈ ℝ*) → (𝑛 ∈ (𝑍(,)𝑊) ↔ (𝑛 ∈ ℝ ∧ 𝑍 < 𝑛𝑛 < 𝑊)))
3929, 37, 38syl2anc 691 . . . . . . . . 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 1238 . . . . . . . 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 679 . . . . . . . . 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 800 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛)
4342adantr 480 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛)
4414ad2antrr 758 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝑍(,)𝑊))
4544ffvelrnda 6267 . . . . . . . . . . . . . . 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 3566 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ ℝ*)
47 simplr 788 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → 𝑦 ∈ (𝑋[,]𝑌))
4844, 47ffvelrnd 6268 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ (𝑍(,)𝑊))
4912, 48sseldi 3566 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ*)
5049adantr 480 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ*)
5126ad2antrl 760 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) → 𝑛 ∈ ℝ)
5251adantr 480 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑛 ∈ ℝ)
5352rexrd 9968 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛𝑛 < 𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑛 ∈ ℝ*)
54 xrlelttr 11863 . . . . . . . . . . . . . 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 1318 . . . . . . . . . . . . 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 706 . . . . . . . . . . . 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 2945 . . . . . . . . . . 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 444 . . . . . . . . . 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 842 . . . . . . . . 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 489 . . . . . . . 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 553 . . . . . . 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 449 . . . . . 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 2997 . . . . 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 3014 . . 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 480 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) → 𝑍 ∈ ℝ*)
6714adantr 480 . . . . . . . 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 790 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) → 𝑦 ∈ (𝑋[,]𝑌))
6967, 68ffvelrnd 6268 . . . . . . 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 3566 . . . . . 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 474 . . . . . 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 11905 . . . . . 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 1318 . . . . 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 11669 . . . . . . . . . 10 (𝑚 ∈ ℚ → 𝑚 ∈ ℝ)
7675ad2antrl 760 . . . . . . . . 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 800 . . . . . . . . 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 9968 . . . . . . . . . 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 480 . . . . . . . . . 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 758 . . . . . . . . . 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 801 . . . . . . . . . 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 478 . . . . . . . . . . 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 480 . . . . . . . . . 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 11866 . . . . . . . . 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 758 . . . . . . . . . 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 12087 . . . . . . . . . 10 ((𝑍 ∈ ℝ*𝑊 ∈ ℝ*) → (𝑚 ∈ (𝑍(,)𝑊) ↔ (𝑚 ∈ ℝ ∧ 𝑍 < 𝑚𝑚 < 𝑊)))
8785, 80, 86syl2anc 691 . . . . . . . . 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 1238 . . . . . . . 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 679 . . . . . . . . 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 801 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))
9190adantr 480 . . . . . . . . . . . . 13 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))
9275ad2antrl 760 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑚 ∈ ℝ)
9392adantr 480 . . . . . . . . . . . . . . 15 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑚 ∈ ℝ)
9493rexrd 9968 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑚 ∈ ℝ*)
9514ad2antrr 758 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝑍(,)𝑊))
96 simplr 788 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑦 ∈ (𝑋[,]𝑌))
9795, 96ffvelrnd 6268 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ (𝑍(,)𝑊))
9812, 97sseldi 3566 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ*)
9998adantr 480 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ*)
10095ffvelrnda 6267 . . . . . . . . . . . . . . 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 3566 . . . . . . . . . . . . . 14 ((((𝜑𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ ℝ*)
102 xrltletr 11864 . . . . . . . . . . . . . 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 1318 . . . . . . . . . . . . 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 707 . . . . . . . . . . . 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 2945 . . . . . . . . . . 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 444 . . . . . . . . . 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 842 . . . . . . . . 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 489 . . . . . . . 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 553 . . . . . . 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 449 . . . . . 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 2997 . . . . 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 3014 . . 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 465 . . . . 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 3086 . . . . 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 266 . . . 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 3046 . . . . . 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 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝑍(,)𝑊))
119118ffvelrnda 6267 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑍(,)𝑊))
1204, 119sseldi 3566 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ ℝ)
1211203biant1d 1433 . . . . . . . . 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 796 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑚 ∈ (𝑍(,)𝑊))
12312, 122sseldi 3566 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑚 ∈ ℝ*)
124 simplrr 797 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑛 ∈ (𝑍(,)𝑊))
12512, 124sseldi 3566 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑛 ∈ ℝ*)
126 elioo2 12087 . . . . . . . . . 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 691 . . . . . . . . 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 270 . . . . . . . 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 2968 . . . . . . 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 6111 . . . . . . . . . . . 12 𝑥((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧)
131130nfel1 2765 . . . . . . . . . . 11 𝑥((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑚(,)𝑛)
132 nfv 1830 . . . . . . . . . . 11 𝑧((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) ∈ (𝑚(,)𝑛)
133 fveq2 6103 . . . . . . . . . . . 12 (𝑧 = 𝑥 → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) = ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥))
134133eleq1d 2672 . . . . . . . . . . 11 (𝑧 = 𝑥 → (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑚(,)𝑛) ↔ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) ∈ (𝑚(,)𝑛)))
135131, 132, 134cbvral 3143 . . . . . . . . . 10 (∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑚(,)𝑛) ↔ ∀𝑥 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) ∈ (𝑚(,)𝑛))
136 simpr 476 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝑋[,]𝑌)) → 𝑥 ∈ (𝑋[,]𝑌))
137 eqid 2610 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) = (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)
138137fmpt 6289 . . . . . . . . . . . . . . 15 (∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑍(,)𝑊) ↔ (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝑍(,)𝑊))
13914, 138sylibr 223 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑍(,)𝑊))
140139r19.21bi 2916 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝑋[,]𝑌)) → 𝐴 ∈ (𝑍(,)𝑊))
141137fvmpt2 6200 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑋[,]𝑌) ∧ 𝐴 ∈ (𝑍(,)𝑊)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) = 𝐴)
142136, 140, 141syl2anc 691 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) = 𝐴)
143142eleq1d 2672 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝑋[,]𝑌)) → (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) ∈ (𝑚(,)𝑛) ↔ 𝐴 ∈ (𝑚(,)𝑛)))
144143ralbidva 2968 . . . . . . . . . 10 (𝜑 → (∀𝑥 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) ∈ (𝑚(,)𝑛) ↔ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛)))
145135, 144syl5bb 271 . . . . . . . . 9 (𝜑 → (∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑚(,)𝑛) ↔ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛)))
146145adantr 480 . . . . . . . 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 480 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑋 ∈ ℝ)
1482adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑌 ∈ ℝ)
1493adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑋𝑌)
15028adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑍 ∈ ℝ*)
15119adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑊 ∈ ℝ*)
152 nfcv 2751 . . . . . . . . . . . . . 14 𝑦𝐴
153 nfcsb1v 3515 . . . . . . . . . . . . . 14 𝑥𝑦 / 𝑥𝐴
154 csbeq1a 3508 . . . . . . . . . . . . . 14 (𝑥 = 𝑦𝐴 = 𝑦 / 𝑥𝐴)
155152, 153, 154cbvmpt 4677 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) = (𝑦 ∈ (𝑋[,]𝑌) ↦ 𝑦 / 𝑥𝐴)
156155, 8syl5eqelr 2693 . . . . . . . . . . . 12 (𝜑 → (𝑦 ∈ (𝑋[,]𝑌) ↦ 𝑦 / 𝑥𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝑍(,)𝑊)))
157156adantr 480 . . . . . . . . . . 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 2751 . . . . . . . . . . . . . 14 𝑦𝐵
159 nfcsb1v 3515 . . . . . . . . . . . . . 14 𝑥𝑦 / 𝑥𝐵
160 csbeq1a 3508 . . . . . . . . . . . . . 14 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
161158, 159, 160cbvmpt 4677 . . . . . . . . . . . . 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 2693 . . . . . . . . . . . 12 (𝜑 → (𝑦 ∈ (𝑋(,)𝑌) ↦ 𝑦 / 𝑥𝐵) ∈ (((𝑋(,)𝑌)–cn→ℂ) ∩ 𝐿1))
164163adantr 480 . . . . . . . . . . 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 2751 . . . . . . . . . . . . . 14 𝑣𝐶
166 nfcsb1v 3515 . . . . . . . . . . . . . 14 𝑢𝑣 / 𝑢𝐶
167 csbeq1a 3508 . . . . . . . . . . . . . 14 (𝑢 = 𝑣𝐶 = 𝑣 / 𝑢𝐶)
168165, 166, 167cbvmpt 4677 . . . . . . . . . . . . 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 2693 . . . . . . . . . . . 12 (𝜑 → (𝑣 ∈ (𝑍(,)𝑊) ↦ 𝑣 / 𝑢𝐶) ∈ ((𝑍(,)𝑊)–cn→ℂ))
171170adantr 480 . . . . . . . . . . 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 6560 . . . . . . . . . . . . 13 (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (ℝ D (𝑦 ∈ (𝑋[,]𝑌) ↦ 𝑦 / 𝑥𝐴))
174172, 173, 1613eqtr3g 2667 . . . . . . . . . . . 12 (𝜑 → (ℝ D (𝑦 ∈ (𝑋[,]𝑌) ↦ 𝑦 / 𝑥𝐴)) = (𝑦 ∈ (𝑋(,)𝑌) ↦ 𝑦 / 𝑥𝐵))
175174adantr 480 . . . . . . . . . . 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 3502 . . . . . . . . . . 11 (𝑣 = 𝑦 / 𝑥𝐴𝑣 / 𝑢𝐶 = 𝑦 / 𝑥𝐴 / 𝑢𝐶)
177 csbeq1 3502 . . . . . . . . . . 11 (𝑦 = 𝑋𝑦 / 𝑥𝐴 = 𝑋 / 𝑥𝐴)
178 csbeq1 3502 . . . . . . . . . . 11 (𝑦 = 𝑌𝑦 / 𝑥𝐴 = 𝑌 / 𝑥𝐴)
179 simprll 798 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑚 ∈ (𝑍(,)𝑊))
180 simprlr 799 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑛 ∈ (𝑍(,)𝑊))
181 simprr 792 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))
182153nfel1 2765 . . . . . . . . . . . . 13 𝑥𝑦 / 𝑥𝐴 ∈ (𝑚(,)𝑛)
183154eleq1d 2672 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝐴 ∈ (𝑚(,)𝑛) ↔ 𝑦 / 𝑥𝐴 ∈ (𝑚(,)𝑛)))
184182, 183rspc 3276 . . . . . . . . . . . 12 (𝑦 ∈ (𝑋[,]𝑌) → (∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛) → 𝑦 / 𝑥𝐴 ∈ (𝑚(,)𝑛)))
185181, 184mpan9 485 . . . . . . . . . . 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 23615 . . . . . . . . . 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 23424 . . . . . . . . . . . 12 ⨜[𝑋 / 𝑥𝐴𝑌 / 𝑥𝐴]𝐶 d𝑢 = ⨜[𝑋 / 𝑥𝐴𝑌 / 𝑥𝐴]𝑣 / 𝑢𝐶 d𝑣
188 nfcvd 2752 . . . . . . . . . . . . . . 15 (𝑋 ∈ ℝ → 𝑥𝐾)
189 itgsubst.k . . . . . . . . . . . . . . 15 (𝑥 = 𝑋𝐴 = 𝐾)
190188, 189csbiegf 3523 . . . . . . . . . . . . . 14 (𝑋 ∈ ℝ → 𝑋 / 𝑥𝐴 = 𝐾)
191 ditgeq1 23418 . . . . . . . . . . . . . 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 2752 . . . . . . . . . . . . . . 15 (𝑌 ∈ ℝ → 𝑥𝐿)
194 itgsubst.l . . . . . . . . . . . . . . 15 (𝑥 = 𝑌𝐴 = 𝐿)
195193, 194csbiegf 3523 . . . . . . . . . . . . . 14 (𝑌 ∈ ℝ → 𝑌 / 𝑥𝐴 = 𝐿)
196 ditgeq2 23419 . . . . . . . . . . . . . 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 2644 . . . . . . . . . . . 12 (𝜑 → ⨜[𝑋 / 𝑥𝐴𝑌 / 𝑥𝐴]𝐶 d𝑢 = ⨜[𝐾𝐿]𝐶 d𝑢)
199187, 198syl5eqr 2658 . . . . . . . . . . 11 (𝜑 → ⨜[𝑋 / 𝑥𝐴𝑌 / 𝑥𝐴]𝑣 / 𝑢𝐶 d𝑣 = ⨜[𝐾𝐿]𝐶 d𝑢)
200199adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → ⨜[𝑋 / 𝑥𝐴𝑌 / 𝑥𝐴]𝑣 / 𝑢𝐶 d𝑣 = ⨜[𝐾𝐿]𝐶 d𝑢)
201154csbeq1d 3506 . . . . . . . . . . . . . 14 (𝑥 = 𝑦𝐴 / 𝑢𝐶 = 𝑦 / 𝑥𝐴 / 𝑢𝐶)
202201, 160oveq12d 6567 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝐴 / 𝑢𝐶 · 𝐵) = (𝑦 / 𝑥𝐴 / 𝑢𝐶 · 𝑦 / 𝑥𝐵))
203 nfcv 2751 . . . . . . . . . . . . 13 𝑦(𝐴 / 𝑢𝐶 · 𝐵)
204 nfcv 2751 . . . . . . . . . . . . . . 15 𝑥𝐶
205153, 204nfcsb 3517 . . . . . . . . . . . . . 14 𝑥𝑦 / 𝑥𝐴 / 𝑢𝐶
206 nfcv 2751 . . . . . . . . . . . . . 14 𝑥 ·
207205, 206, 159nfov 6575 . . . . . . . . . . . . 13 𝑥(𝑦 / 𝑥𝐴 / 𝑢𝐶 · 𝑦 / 𝑥𝐵)
208202, 203, 207cbvditg 23424 . . . . . . . . . . . 12 ⨜[𝑋𝑌](𝐴 / 𝑢𝐶 · 𝐵) d𝑥 = ⨜[𝑋𝑌](𝑦 / 𝑥𝐴 / 𝑢𝐶 · 𝑦 / 𝑥𝐵) d𝑦
209 ioossicc 12130 . . . . . . . . . . . . . . . . . 18 (𝑋(,)𝑌) ⊆ (𝑋[,]𝑌)
210209sseli 3564 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝑋(,)𝑌) → 𝑥 ∈ (𝑋[,]𝑌))
211210, 140sylan2 490 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ (𝑍(,)𝑊))
212 nfcvd 2752 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ (𝑍(,)𝑊) → 𝑢𝐸)
213 itgsubst.e . . . . . . . . . . . . . . . . 17 (𝑢 = 𝐴𝐶 = 𝐸)
214212, 213csbiegf 3523 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (𝑍(,)𝑊) → 𝐴 / 𝑢𝐶 = 𝐸)
215211, 214syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 / 𝑢𝐶 = 𝐸)
216215oveq1d 6564 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → (𝐴 / 𝑢𝐶 · 𝐵) = (𝐸 · 𝐵))
217216itgeq2dv 23354 . . . . . . . . . . . . 13 (𝜑 → ∫(𝑋(,)𝑌)(𝐴 / 𝑢𝐶 · 𝐵) d𝑥 = ∫(𝑋(,)𝑌)(𝐸 · 𝐵) d𝑥)
2183ditgpos 23426 . . . . . . . . . . . . 13 (𝜑 → ⨜[𝑋𝑌](𝐴 / 𝑢𝐶 · 𝐵) d𝑥 = ∫(𝑋(,)𝑌)(𝐴 / 𝑢𝐶 · 𝐵) d𝑥)
2193ditgpos 23426 . . . . . . . . . . . . 13 (𝜑 → ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥 = ∫(𝑋(,)𝑌)(𝐸 · 𝐵) d𝑥)
220217, 218, 2193eqtr4d 2654 . . . . . . . . . . . 12 (𝜑 → ⨜[𝑋𝑌](𝐴 / 𝑢𝐶 · 𝐵) d𝑥 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥)
221208, 220syl5eqr 2658 . . . . . . . . . . 11 (𝜑 → ⨜[𝑋𝑌](𝑦 / 𝑥𝐴 / 𝑢𝐶 · 𝑦 / 𝑥𝐵) d𝑦 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥)
222221adantr 480 . . . . . . . . . 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 2652 . . . . . . . . 9 ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → ⨜[𝐾𝐿]𝐶 d𝑢 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥)
224223expr 641 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) → (∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛) → ⨜[𝐾𝐿]𝐶 d𝑢 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥))
225146, 224sylbid 229 . . . . . . 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 229 . . . . . 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 232 . . . . 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 3020 . . . 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 231 . . 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 499 . 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ⦋csb 3499   ∩ cin 3539   ⊆ wss 3540   class class class wbr 4583   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ℝcr 9814   · cmul 9820  ℝ*cxr 9952   < clt 9953   ≤ cle 9954  ℚcq 11664  (,)cioo 12046  [,]cicc 12049  –cn→ccncf 22487  𝐿1cibl 23192  ∫citg 23193  ⨜cdit 23416   D cdv 23433 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cc 9140  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ioc 12051  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068  df-sum 14265  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lp 20750  df-perf 20751  df-cn 20841  df-cnp 20842  df-haus 20929  df-cmp 21000  df-tx 21175  df-hmeo 21368  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-xms 21935  df-ms 21936  df-tms 21937  df-cncf 22489  df-ovol 23040  df-vol 23041  df-mbf 23194  df-itg1 23195  df-itg2 23196  df-ibl 23197  df-itg 23198  df-0p 23243  df-ditg 23417  df-limc 23436  df-dv 23437 This theorem is referenced by:  itgsubsticclem  38867
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