itgsubst - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > itgsubst		Structured version Visualization version GIF version

Theorem itgsubst 26016

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 26015, 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	⊢ (𝜑 → 𝑋 ∈ ℝ)
itgsubst.y	⊢ (𝜑 → 𝑌 ∈ ℝ)
itgsubst.le	⊢ (𝜑 → 𝑋 ≤ 𝑌)
itgsubst.z	⊢ (𝜑 → 𝑍 ∈ ℝ^*)
itgsubst.w	⊢ (𝜑 → 𝑊 ∈ 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 = 𝐴 → 𝐶 = 𝐸)
itgsubst.k	⊢ (𝑥 = 𝑋 → 𝐴 = 𝐾)
itgsubst.l	⊢ (𝑥 = 𝑌 → 𝐴 = 𝐿)

**Assertion**
Ref	Expression
itgsubst	⊢ (𝜑 → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](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.

Step	Hyp	Ref	Expression
1		itgsubst.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ)
2		itgsubst.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ ℝ)
3		itgsubst.le	. . 3 ⊢ (𝜑 → 𝑋 ≤ 𝑌)
4		ioossre 13360	. . . . 5 ⊢ (𝑍(,)𝑊) ⊆ ℝ
5		ax-resscn 11095	. . . . 5 ⊢ ℝ ⊆ ℂ
6		cncfss 24866	. . . . 5 ⊢ (((𝑍(,)𝑊) ⊆ ℝ ∧ ℝ ⊆ ℂ) → ((𝑋[,]𝑌)–cn→(𝑍(,)𝑊)) ⊆ ((𝑋[,]𝑌)–cn→ℝ))
7	4, 5, 6	mp2an 693	. . . 4 ⊢ ((𝑋[,]𝑌)–cn→(𝑍(,)𝑊)) ⊆ ((𝑋[,]𝑌)–cn→ℝ)
8		itgsubst.a	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝑍(,)𝑊)))
9	7, 8	sselid 3920	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→ℝ))
10	1, 2, 3, 9	evthicc 25426	. 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 11189	. . . . . . . 8 ⊢ ℝ ⊆ ℝ^*
12	4, 11	sstri 3932	. . . . . . 7 ⊢ (𝑍(,)𝑊) ⊆ ℝ^*
13		cncff 24860	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝑍(,)𝑊)) → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝑍(,)𝑊))
14	8, 13	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝑍(,)𝑊))
15	14	adantr 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 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) → 𝑦 ∈ (𝑋[,]𝑌))
17	15, 16	ffvelcdmd 7038	. . . . . . 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))
18	12, 17	sselid 3920	. . . . . 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 ⊢ (𝜑 → 𝑊 ∈ ℝ^*)
20	19	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) → 𝑊 ∈ ℝ^*)
21		eliooord 13358	. . . . . . . 8 ⊢ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ (𝑍(,)𝑊) → (𝑍 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∧ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑊))
22	17, 21	syl 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))
23	22	simprd 495	. . . . . 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 13152	. . . . . 6 ⊢ ((((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ^* ∧ 𝑊 ∈ ℝ^* ∧ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑊) → ∃𝑛 ∈ ℚ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛 ∧ 𝑛 < 𝑊))
25	18, 20, 23, 24	syl3anc 1374	. . . . 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) < 𝑛 ∧ 𝑛 < 𝑊))
26		qre 12903	. . . . . . 7 ⊢ (𝑛 ∈ ℚ → 𝑛 ∈ ℝ)
27	26	ad2antrl 729	. . . . . 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) < 𝑛 ∧ 𝑛 < 𝑊))) → 𝑛 ∈ ℝ)
28		itgsubst.z	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ ℝ^*)
29	28	ad2antrr 727	. . . . . . 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) < 𝑛 ∧ 𝑛 < 𝑊))) → 𝑍 ∈ ℝ^*)
30	18	adantr 480	. . . . . . 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) < 𝑛 ∧ 𝑛 < 𝑊))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ^*)
31	27	rexrd 11195	. . . . . . 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) < 𝑛 ∧ 𝑛 < 𝑊))) → 𝑛 ∈ ℝ^*)
32	22	simpld 494	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) → 𝑍 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))
33	32	adantr 480	. . . . . . 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) < 𝑛 ∧ 𝑛 < 𝑊))) → 𝑍 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))
34		simprrl 781	. . . . . . 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) < 𝑛 ∧ 𝑛 < 𝑊))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛)
35	29, 30, 31, 33, 34	xrlttrd 13110	. . . . . 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) < 𝑛 ∧ 𝑛 < 𝑊))) → 𝑍 < 𝑛)
36		simprrr 782	. . . . . 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) < 𝑛 ∧ 𝑛 < 𝑊))) → 𝑛 < 𝑊)
37	19	ad2antrr 727	. . . . . . 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) < 𝑛 ∧ 𝑛 < 𝑊))) → 𝑊 ∈ ℝ^*)
38		elioo2 13339	. . . . . . 7 ⊢ ((𝑍 ∈ ℝ^* ∧ 𝑊 ∈ ℝ^*) → (𝑛 ∈ (𝑍(,)𝑊) ↔ (𝑛 ∈ ℝ ∧ 𝑍 < 𝑛 ∧ 𝑛 < 𝑊)))
39	29, 37, 38	syl2anc 585	. . . . . 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) < 𝑛 ∧ 𝑛 < 𝑊))) → (𝑛 ∈ (𝑍(,)𝑊) ↔ (𝑛 ∈ ℝ ∧ 𝑍 < 𝑛 ∧ 𝑛 < 𝑊)))
40	27, 35, 36, 39	mpbir3and 1344	. . . . 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) < 𝑛 ∧ 𝑛 < 𝑊))) → 𝑛 ∈ (𝑍(,)𝑊))
41		anass 468	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (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 781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛 ∧ 𝑛 < 𝑊))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛)
43	42	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛 ∧ 𝑛 < 𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛)
44	14	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛 ∧ 𝑛 < 𝑊))) → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝑍(,)𝑊))
45	44	ffvelcdmda 7037	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛 ∧ 𝑛 < 𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑍(,)𝑊))
46	12, 45	sselid 3920	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛 ∧ 𝑛 < 𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ ℝ^*)
47		simplr 769	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛 ∧ 𝑛 < 𝑊))) → 𝑦 ∈ (𝑋[,]𝑌))
48	44, 47	ffvelcdmd 7038	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛 ∧ 𝑛 < 𝑊))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ (𝑍(,)𝑊))
49	12, 48	sselid 3920	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛 ∧ 𝑛 < 𝑊))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ^*)
50	49	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛 ∧ 𝑛 < 𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ^*)
51	26	ad2antrl 729	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛 ∧ 𝑛 < 𝑊))) → 𝑛 ∈ ℝ)
52	51	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛 ∧ 𝑛 < 𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑛 ∈ ℝ)
53	52	rexrd 11195	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛 ∧ 𝑛 < 𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑛 ∈ ℝ^*)
54		xrlelttr 13107	. . . . . . . . . . 11 ⊢ ((((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))
55	46, 50, 53, 54	syl3anc 1374	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛 ∧ 𝑛 < 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))
56	43, 55	mpan2d 695	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛 ∧ 𝑛 < 𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛))
57	56	ralimdva 3150	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛 ∧ 𝑛 < 𝑊))) → (∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) → ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛))
58	57	imp 406	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛 ∧ 𝑛 < 𝑊))) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)) → ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛)
59	58	an32s 653	. . . . . 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) < 𝑛 ∧ 𝑛 < 𝑊))) → ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛)
60	41, 59	sylanbr 583	. . . . 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) < 𝑛 ∧ 𝑛 < 𝑊))) → ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛)
61	25, 40, 60	reximssdv 3156	. . . 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)
62	61	rexlimdvaa 3140	. . 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))
63	28	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) → 𝑍 ∈ ℝ^*)
64	14	adantr 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))
65		simprl 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) → 𝑦 ∈ (𝑋[,]𝑌))
66	64, 65	ffvelcdmd 7038	. . . . . . 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))
67	12, 66	sselid 3920	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ^*)
68	66, 21	syl 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))
69	68	simpld 494	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) → 𝑍 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))
70		qbtwnxr 13152	. . . . . 6 ⊢ ((𝑍 ∈ ℝ^* ∧ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ^* ∧ 𝑍 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)) → ∃𝑚 ∈ ℚ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))
71	63, 67, 69, 70	syl3anc 1374	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) → ∃𝑚 ∈ ℚ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))
72		qre 12903	. . . . . . 7 ⊢ (𝑚 ∈ ℚ → 𝑚 ∈ ℝ)
73	72	ad2antrl 729	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑚 ∈ ℝ)
74		simprrl 781	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑍 < 𝑚)
75	73	rexrd 11195	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑚 ∈ ℝ^*)
76	67	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ^*)
77	19	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑊 ∈ ℝ^*)
78		simprrr 782	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))
79	68	simprd 495	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑊)
80	79	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑊)
81	75, 76, 77, 78, 80	xrlttrd 13110	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑚 < 𝑊)
82	28	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑍 ∈ ℝ^*)
83		elioo2 13339	. . . . . . 7 ⊢ ((𝑍 ∈ ℝ^* ∧ 𝑊 ∈ ℝ^*) → (𝑚 ∈ (𝑍(,)𝑊) ↔ (𝑚 ∈ ℝ ∧ 𝑍 < 𝑚 ∧ 𝑚 < 𝑊)))
84	82, 77, 83	syl2anc 585	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → (𝑚 ∈ (𝑍(,)𝑊) ↔ (𝑚 ∈ ℝ ∧ 𝑍 < 𝑚 ∧ 𝑚 < 𝑊)))
85	73, 74, 81, 84	mpbir3and 1344	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑚 ∈ (𝑍(,)𝑊))
86		anass 468	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (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))))
87		simprrr 782	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))
88	87	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))
89	72	ad2antrl 729	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑚 ∈ ℝ)
90	89	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑚 ∈ ℝ)
91	90	rexrd 11195	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑚 ∈ ℝ^*)
92	14	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝑍(,)𝑊))
93		simplr 769	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑦 ∈ (𝑋[,]𝑌))
94	92, 93	ffvelcdmd 7038	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ (𝑍(,)𝑊))
95	12, 94	sselid 3920	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ^*)
96	95	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ^*)
97	92	ffvelcdmda 7037	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑍(,)𝑊))
98	12, 97	sselid 3920	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ ℝ^*)
99		xrltletr 13108	. . . . . . . . . . 11 ⊢ ((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)))
100	91, 96, 98, 99	syl3anc 1374	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((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)))
101	88, 100	mpand 696	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) → 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧)))
102	101	ralimdva 3150	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((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)))
103	102	imp 406	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((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))
104	103	an32s 653	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → ∀𝑧 ∈ (𝑋[,]𝑌)𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))
105	86, 104	sylanbr 583	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → ∀𝑧 ∈ (𝑋[,]𝑌)𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))
106	71, 85, 105	reximssdv 3156	. . . 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))
107	106	rexlimdvaa 3140	. . 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)))
108		ancom 460	. . . . 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))
109		reeanv 3210	. . . . 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))
110	108, 109	bitr4i 278	. . . 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))
111		r19.26 3098	. . . . . 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))
112	14	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝑍(,)𝑊))
113	112	ffvelcdmda 7037	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑍(,)𝑊))
114	4, 113	sselid 3920	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ ℝ)
115	114	3biant1d 1481	. . . . . . . . 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)))
116		simplrl 777	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑚 ∈ (𝑍(,)𝑊))
117	12, 116	sselid 3920	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑚 ∈ ℝ^*)
118		simplrr 778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑛 ∈ (𝑍(,)𝑊))
119	12, 118	sselid 3920	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑛 ∈ ℝ^*)
120		elioo2 13339	. . . . . . . . . 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)))
121	117, 119, 120	syl2anc 585	. . . . . . . . 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)))
122	115, 121	bitr4d 282	. . . . . . . 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)))
123	122	ralbidva 3159	. . . . . . 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)))
124		nffvmpt1 6852	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧)
125	124	nfel1 2916	. . . . . . . . . . 11 ⊢ Ⅎ𝑥((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑚(,)𝑛)
126		nfv 1916	. . . . . . . . . . 11 ⊢ Ⅎ𝑧((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) ∈ (𝑚(,)𝑛)
127		fveq2 6841	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) = ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥))
128	127	eleq1d 2822	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑚(,)𝑛) ↔ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) ∈ (𝑚(,)𝑛)))
129	125, 126, 128	cbvralw 3280	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑚(,)𝑛) ↔ ∀𝑥 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) ∈ (𝑚(,)𝑛))
130		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋[,]𝑌)) → 𝑥 ∈ (𝑋[,]𝑌))
131	14	fvmptelcdm 7066	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋[,]𝑌)) → 𝐴 ∈ (𝑍(,)𝑊))
132		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) = (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)
133	132	fvmpt2 6960	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝑋[,]𝑌) ∧ 𝐴 ∈ (𝑍(,)𝑊)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) = 𝐴)
134	130, 131, 133	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) = 𝐴)
135	134	eleq1d 2822	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋[,]𝑌)) → (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) ∈ (𝑚(,)𝑛) ↔ 𝐴 ∈ (𝑚(,)𝑛)))
136	135	ralbidva 3159	. . . . . . . . . 10 ⊢ (𝜑 → (∀𝑥 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) ∈ (𝑚(,)𝑛) ↔ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛)))
137	129, 136	bitrid 283	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑚(,)𝑛) ↔ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛)))
138	137	adantr 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)))
139	1	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑋 ∈ ℝ)
140	2	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑌 ∈ ℝ)
141	3	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑋 ≤ 𝑌)
142	28	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑍 ∈ ℝ^*)
143	19	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑊 ∈ ℝ^*)
144		nfcv 2899	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦𝐴
145		nfcsb1v 3862	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴
146		csbeq1a 3852	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴)
147	144, 145, 146	cbvmpt 5188	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) = (𝑦 ∈ (𝑋[,]𝑌) ↦ ⦋𝑦 / 𝑥⦌𝐴)
148	147, 8	eqeltrrid 2842	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑦 ∈ (𝑋[,]𝑌) ↦ ⦋𝑦 / 𝑥⦌𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝑍(,)𝑊)))
149	148	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → (𝑦 ∈ (𝑋[,]𝑌) ↦ ⦋𝑦 / 𝑥⦌𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝑍(,)𝑊)))
150		nfcv 2899	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦𝐵
151		nfcsb1v 3862	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
152		csbeq1a 3852	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
153	150, 151, 152	cbvmpt 5188	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) = (𝑦 ∈ (𝑋(,)𝑌) ↦ ⦋𝑦 / 𝑥⦌𝐵)
154		itgsubst.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ (((𝑋(,)𝑌)–cn→ℂ) ∩ 𝐿¹))
155	153, 154	eqeltrrid 2842	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑦 ∈ (𝑋(,)𝑌) ↦ ⦋𝑦 / 𝑥⦌𝐵) ∈ (((𝑋(,)𝑌)–cn→ℂ) ∩ 𝐿¹))
156	155	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → (𝑦 ∈ (𝑋(,)𝑌) ↦ ⦋𝑦 / 𝑥⦌𝐵) ∈ (((𝑋(,)𝑌)–cn→ℂ) ∩ 𝐿¹))
157		nfcv 2899	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑣𝐶
158		nfcsb1v 3862	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑢⦋𝑣 / 𝑢⦌𝐶
159		csbeq1a 3852	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑣 → 𝐶 = ⦋𝑣 / 𝑢⦌𝐶)
160	157, 158, 159	cbvmpt 5188	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ (𝑍(,)𝑊) ↦ 𝐶) = (𝑣 ∈ (𝑍(,)𝑊) ↦ ⦋𝑣 / 𝑢⦌𝐶)
161		itgsubst.c	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑢 ∈ (𝑍(,)𝑊) ↦ 𝐶) ∈ ((𝑍(,)𝑊)–cn→ℂ))
162	160, 161	eqeltrrid 2842	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑣 ∈ (𝑍(,)𝑊) ↦ ⦋𝑣 / 𝑢⦌𝐶) ∈ ((𝑍(,)𝑊)–cn→ℂ))
163	162	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → (𝑣 ∈ (𝑍(,)𝑊) ↦ ⦋𝑣 / 𝑢⦌𝐶) ∈ ((𝑍(,)𝑊)–cn→ℂ))
164		itgsubst.da	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))
165	147	oveq2i 7378	. . . . . . . . . . . . 13 ⊢ (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (ℝ D (𝑦 ∈ (𝑋[,]𝑌) ↦ ⦋𝑦 / 𝑥⦌𝐴))
166	164, 165, 153	3eqtr3g 2795	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝑋[,]𝑌) ↦ ⦋𝑦 / 𝑥⦌𝐴)) = (𝑦 ∈ (𝑋(,)𝑌) ↦ ⦋𝑦 / 𝑥⦌𝐵))
167	166	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → (ℝ D (𝑦 ∈ (𝑋[,]𝑌) ↦ ⦋𝑦 / 𝑥⦌𝐴)) = (𝑦 ∈ (𝑋(,)𝑌) ↦ ⦋𝑦 / 𝑥⦌𝐵))
168		csbeq1 3841	. . . . . . . . . . 11 ⊢ (𝑣 = ⦋𝑦 / 𝑥⦌𝐴 → ⦋𝑣 / 𝑢⦌𝐶 = ⦋⦋𝑦 / 𝑥⦌𝐴 / 𝑢⦌𝐶)
169		csbeq1 3841	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑋 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑋 / 𝑥⦌𝐴)
170		csbeq1 3841	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑌 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑌 / 𝑥⦌𝐴)
171		simprll 779	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑚 ∈ (𝑍(,)𝑊))
172		simprlr 780	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑛 ∈ (𝑍(,)𝑊))
173		simprr 773	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))
174	145	nfel1 2916	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 ∈ (𝑚(,)𝑛)
175	146	eleq1d 2822	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ (𝑚(,)𝑛) ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ (𝑚(,)𝑛)))
176	174, 175	rspc 3553	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝑋[,]𝑌) → (∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛) → ⦋𝑦 / 𝑥⦌𝐴 ∈ (𝑚(,)𝑛)))
177	173, 176	mpan9 506	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) ∧ 𝑦 ∈ (𝑋[,]𝑌)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ (𝑚(,)𝑛))
178	139, 140, 141, 142, 143, 149, 156, 163, 167, 168, 169, 170, 171, 172, 177	itgsubstlem 26015	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → ⨜[⦋𝑋 / 𝑥⦌𝐴 → ⦋𝑌 / 𝑥⦌𝐴]⦋𝑣 / 𝑢⦌𝐶 d𝑣 = ⨜[𝑋 → 𝑌](⦋⦋𝑦 / 𝑥⦌𝐴 / 𝑢⦌𝐶 · ⦋𝑦 / 𝑥⦌𝐵) d𝑦)
179	159, 157, 158	cbvditg 25821	. . . . . . . . . . . 12 ⊢ ⨜[⦋𝑋 / 𝑥⦌𝐴 → ⦋𝑌 / 𝑥⦌𝐴]𝐶 d𝑢 = ⨜[⦋𝑋 / 𝑥⦌𝐴 → ⦋𝑌 / 𝑥⦌𝐴]⦋𝑣 / 𝑢⦌𝐶 d𝑣
180		nfcvd 2900	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ ℝ → Ⅎ𝑥𝐾)
181		itgsubst.k	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑋 → 𝐴 = 𝐾)
182	180, 181	csbiegf 3871	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℝ → ⦋𝑋 / 𝑥⦌𝐴 = 𝐾)
183		ditgeq1 25815	. . . . . . . . . . . . . 14 ⊢ (⦋𝑋 / 𝑥⦌𝐴 = 𝐾 → ⨜[⦋𝑋 / 𝑥⦌𝐴 → ⦋𝑌 / 𝑥⦌𝐴]𝐶 d𝑢 = ⨜[𝐾 → ⦋𝑌 / 𝑥⦌𝐴]𝐶 d𝑢)
184	1, 182, 183	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ⨜[⦋𝑋 / 𝑥⦌𝐴 → ⦋𝑌 / 𝑥⦌𝐴]𝐶 d𝑢 = ⨜[𝐾 → ⦋𝑌 / 𝑥⦌𝐴]𝐶 d𝑢)
185		nfcvd 2900	. . . . . . . . . . . . . . 15 ⊢ (𝑌 ∈ ℝ → Ⅎ𝑥𝐿)
186		itgsubst.l	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑌 → 𝐴 = 𝐿)
187	185, 186	csbiegf 3871	. . . . . . . . . . . . . 14 ⊢ (𝑌 ∈ ℝ → ⦋𝑌 / 𝑥⦌𝐴 = 𝐿)
188		ditgeq2 25816	. . . . . . . . . . . . . 14 ⊢ (⦋𝑌 / 𝑥⦌𝐴 = 𝐿 → ⨜[𝐾 → ⦋𝑌 / 𝑥⦌𝐴]𝐶 d𝑢 = ⨜[𝐾 → 𝐿]𝐶 d𝑢)
189	2, 187, 188	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ⨜[𝐾 → ⦋𝑌 / 𝑥⦌𝐴]𝐶 d𝑢 = ⨜[𝐾 → 𝐿]𝐶 d𝑢)
190	184, 189	eqtrd 2772	. . . . . . . . . . . 12 ⊢ (𝜑 → ⨜[⦋𝑋 / 𝑥⦌𝐴 → ⦋𝑌 / 𝑥⦌𝐴]𝐶 d𝑢 = ⨜[𝐾 → 𝐿]𝐶 d𝑢)
191	179, 190	eqtr3id 2786	. . . . . . . . . . 11 ⊢ (𝜑 → ⨜[⦋𝑋 / 𝑥⦌𝐴 → ⦋𝑌 / 𝑥⦌𝐴]⦋𝑣 / 𝑢⦌𝐶 d𝑣 = ⨜[𝐾 → 𝐿]𝐶 d𝑢)
192	191	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → ⨜[⦋𝑋 / 𝑥⦌𝐴 → ⦋𝑌 / 𝑥⦌𝐴]⦋𝑣 / 𝑢⦌𝐶 d𝑣 = ⨜[𝐾 → 𝐿]𝐶 d𝑢)
193	146	csbeq1d 3842	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ⦋𝐴 / 𝑢⦌𝐶 = ⦋⦋𝑦 / 𝑥⦌𝐴 / 𝑢⦌𝐶)
194	193, 152	oveq12d 7385	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (⦋𝐴 / 𝑢⦌𝐶 · 𝐵) = (⦋⦋𝑦 / 𝑥⦌𝐴 / 𝑢⦌𝐶 · ⦋𝑦 / 𝑥⦌𝐵))
195		nfcv 2899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦(⦋𝐴 / 𝑢⦌𝐶 · 𝐵)
196		nfcv 2899	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥𝐶
197	145, 196	nfcsbw 3864	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⦋⦋𝑦 / 𝑥⦌𝐴 / 𝑢⦌𝐶
198		nfcv 2899	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥 ·
199	197, 198, 151	nfov 7397	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(⦋⦋𝑦 / 𝑥⦌𝐴 / 𝑢⦌𝐶 · ⦋𝑦 / 𝑥⦌𝐵)
200	194, 195, 199	cbvditg 25821	. . . . . . . . . . . 12 ⊢ ⨜[𝑋 → 𝑌](⦋𝐴 / 𝑢⦌𝐶 · 𝐵) d𝑥 = ⨜[𝑋 → 𝑌](⦋⦋𝑦 / 𝑥⦌𝐴 / 𝑢⦌𝐶 · ⦋𝑦 / 𝑥⦌𝐵) d𝑦
201		ioossicc 13386	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋(,)𝑌) ⊆ (𝑋[,]𝑌)
202	201	sseli 3918	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝑋(,)𝑌) → 𝑥 ∈ (𝑋[,]𝑌))
203	202, 131	sylan2 594	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ (𝑍(,)𝑊))
204		nfcvd 2900	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ (𝑍(,)𝑊) → Ⅎ𝑢𝐸)
205		itgsubst.e	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝐴 → 𝐶 = 𝐸)
206	204, 205	csbiegf 3871	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ (𝑍(,)𝑊) → ⦋𝐴 / 𝑢⦌𝐶 = 𝐸)
207	203, 206	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → ⦋𝐴 / 𝑢⦌𝐶 = 𝐸)
208	207	oveq1d 7382	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (⦋𝐴 / 𝑢⦌𝐶 · 𝐵) = (𝐸 · 𝐵))
209	208	itgeq2dv 25749	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∫(𝑋(,)𝑌)(⦋𝐴 / 𝑢⦌𝐶 · 𝐵) d𝑥 = ∫(𝑋(,)𝑌)(𝐸 · 𝐵) d𝑥)
210	3	ditgpos 25823	. . . . . . . . . . . . 13 ⊢ (𝜑 → ⨜[𝑋 → 𝑌](⦋𝐴 / 𝑢⦌𝐶 · 𝐵) d𝑥 = ∫(𝑋(,)𝑌)(⦋𝐴 / 𝑢⦌𝐶 · 𝐵) d𝑥)
211	3	ditgpos 25823	. . . . . . . . . . . . 13 ⊢ (𝜑 → ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥 = ∫(𝑋(,)𝑌)(𝐸 · 𝐵) d𝑥)
212	209, 210, 211	3eqtr4d 2782	. . . . . . . . . . . 12 ⊢ (𝜑 → ⨜[𝑋 → 𝑌](⦋𝐴 / 𝑢⦌𝐶 · 𝐵) d𝑥 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥)
213	200, 212	eqtr3id 2786	. . . . . . . . . . 11 ⊢ (𝜑 → ⨜[𝑋 → 𝑌](⦋⦋𝑦 / 𝑥⦌𝐴 / 𝑢⦌𝐶 · ⦋𝑦 / 𝑥⦌𝐵) d𝑦 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥)
214	213	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → ⨜[𝑋 → 𝑌](⦋⦋𝑦 / 𝑥⦌𝐴 / 𝑢⦌𝐶 · ⦋𝑦 / 𝑥⦌𝐵) d𝑦 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥)
215	178, 192, 214	3eqtr3d 2780	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥)
216	215	expr 456	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) → (∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛) → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥))
217	138, 216	sylbid 240	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) → (∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑚(,)𝑛) → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥))
218	123, 217	sylbid 240	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) → (∀𝑧 ∈ (𝑋[,]𝑌)(𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∧ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) < 𝑛) → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥))
219	111, 218	biimtrrid 243	. . . . 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) → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥))
220	219	rexlimdvva 3195	. . . 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) → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥))
221	110, 220	biimtrid 242	. . 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)) → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥))
222	62, 107, 221	syl2and 609	. 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)) → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥))
223	10, 222	mpd 15	1 ⊢ (𝜑 → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 ⦋csb 3838 ∩ cin 3889 ⊆ wss 3890 class class class wbr 5086 ↦ cmpt 5167 ⟶wf 6495 ‘cfv 6499 (class class class)co 7367 ℂcc 11036 ℝcr 11037 · cmul 11043 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 ℚcq 12898 (,)cioo 13298 [,]cicc 13301 –cn→ccncf 24843 𝐿¹cibl 25584 ∫citg 25585 ⨜cdit 25813 D cdv 25830

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7689 ax-inf2 9562 ax-cc 10357 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-symdif 4194 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6266 df-ord 6327 df-on 6328 df-lim 6329 df-suc 6330 df-iota 6455 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-ofr 7632 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-omul 8410 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-acn 9866 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-cmp 23352 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-ovol 25431 df-vol 25432 df-mbf 25586 df-itg1 25587 df-itg2 25588 df-ibl 25589 df-itg 25590 df-0p 25637 df-ditg 25814 df-limc 25833 df-dv 25834

This theorem is referenced by: itgsubsticclem 46403

W3C validator