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Theorem itgsubst 25557

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 25556, 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 13381	. . . . 5 ⊢ (𝑍(,)𝑊) ⊆ ℝ
5		ax-resscn 11163	. . . . 5 ⊢ ℝ ⊆ ℂ
6		cncfss 24406	. . . . 5 ⊢ (((𝑍(,)𝑊) ⊆ ℝ ∧ ℝ ⊆ ℂ) → ((𝑋[,]𝑌)–cn→(𝑍(,)𝑊)) ⊆ ((𝑋[,]𝑌)–cn→ℝ))
7	4, 5, 6	mp2an 690	. . . 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 3979	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→ℝ))
10	1, 2, 3, 9	evthicc 24967	. 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 11254	. . . . . . . 8 ⊢ ℝ ⊆ ℝ^*
12	4, 11	sstri 3990	. . . . . . 7 ⊢ (𝑍(,)𝑊) ⊆ ℝ^*
13		cncff 24400	. . . . . . . . . 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 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 769	. . . . . . . 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 7084	. . . . . . 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 3979	. . . . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))) → 𝑊 ∈ ℝ^*)
21		eliooord 13379	. . . . . . . 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 496	. . . . . 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 13175	. . . . . 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 1371	. . . . 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 12933	. . . . . . 7 ⊢ (𝑛 ∈ ℚ → 𝑛 ∈ ℝ)
27	26	ad2antrl 726	. . . . . 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 724	. . . . . . 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 481	. . . . . . 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 11260	. . . . . . 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 495	. . . . . . . 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 481	. . . . . . 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 779	. . . . . . 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 13134	. . . . . 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 780	. . . . . 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 724	. . . . . . 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 13361	. . . . . . 7 ⊢ ((𝑍 ∈ ℝ^* ∧ 𝑊 ∈ ℝ^*) → (𝑛 ∈ (𝑍(,)𝑊) ↔ (𝑛 ∈ ℝ ∧ 𝑍 < 𝑛 ∧ 𝑛 < 𝑊)))
39	29, 37, 38	syl2anc 584	. . . . . 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 1342	. . . . 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 469	. . . . . 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 779	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛 ∧ 𝑛 < 𝑊))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛)
43	42	adantr 481	. . . . . . . . . 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 724	. . . . . . . . . . . . 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 7083	. . . . . . . . . . . 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 3979	. . . . . . . . . . 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 767	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛 ∧ 𝑛 < 𝑊))) → 𝑦 ∈ (𝑋[,]𝑌))
48	44, 47	ffvelcdmd 7084	. . . . . . . . . . . . 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 3979	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛 ∧ 𝑛 < 𝑊))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ^*)
50	49	adantr 481	. . . . . . . . . . 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 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛 ∧ 𝑛 < 𝑊))) → 𝑛 ∈ ℝ)
52	51	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛 ∧ 𝑛 < 𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑛 ∈ ℝ)
53	52	rexrd 11260	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑛 ∈ ℚ ∧ (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) < 𝑛 ∧ 𝑛 < 𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑛 ∈ ℝ^*)
54		xrlelttr 13131	. . . . . . . . . . 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 1371	. . . . . . . . . 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 692	. . . . . . . . 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 3167	. . . . . . . 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 407	. . . . . . 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 650	. . . . . 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 582	. . . . 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 3172	. . . 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 3156	. . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋[,]𝑌) ∧ ∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ≤ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧))) → 𝑍 ∈ ℝ^*)
64	14	adantr 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))
65		simprl 769	. . . . . . . 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 7084	. . . . . . 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 3979	. . . . . 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 495	. . . . . 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 13175	. . . . . 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 1371	. . . . 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 12933	. . . . . . 7 ⊢ (𝑚 ∈ ℚ → 𝑚 ∈ ℝ)
73	72	ad2antrl 726	. . . . . 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 779	. . . . . 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 11260	. . . . . . 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 481	. . . . . . 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 724	. . . . . . 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 780	. . . . . . 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 496	. . . . . . . 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 481	. . . . . . 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 13134	. . . . . 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 724	. . . . . . 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 13361	. . . . . . 7 ⊢ ((𝑍 ∈ ℝ^* ∧ 𝑊 ∈ ℝ^*) → (𝑚 ∈ (𝑍(,)𝑊) ↔ (𝑚 ∈ ℝ ∧ 𝑍 < 𝑚 ∧ 𝑚 < 𝑊)))
84	82, 77, 83	syl2anc 584	. . . . . 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 1342	. . . . 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 469	. . . . . 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 780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦))
88	87	adantr 481	. . . . . . . . . 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 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑚 ∈ ℝ)
90	89	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑚 ∈ ℝ)
91	90	rexrd 11260	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑚 ∈ ℝ^*)
92	14	ad2antrr 724	. . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → 𝑦 ∈ (𝑋[,]𝑌))
94	92, 93	ffvelcdmd 7084	. . . . . . . . . . . . 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 3979	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑋[,]𝑌)) ∧ (𝑚 ∈ ℚ ∧ (𝑍 < 𝑚 ∧ 𝑚 < ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦)))) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑦) ∈ ℝ^*)
96	95	adantr 481	. . . . . . . . . . 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 7083	. . . . . . . . . . . 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 3979	. . . . . . . . . . 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 13132	. . . . . . . . . . 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 1371	. . . . . . . . . 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 693	. . . . . . . . 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 3167	. . . . . . . 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 407	. . . . . . 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 650	. . . . . 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 582	. . . . 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 3172	. . . 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 3156	. . 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 461	. . . . 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 3226	. . . . 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 277	. . . 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 3111	. . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴):(𝑋[,]𝑌)⟶(𝑍(,)𝑊))
113	112	ffvelcdmda 7083	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑍(,)𝑊))
114	4, 113	sselid 3979	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ ℝ)
115	114	3biant1d 1478	. . . . . . . . 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 775	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑚 ∈ (𝑍(,)𝑊))
117	12, 116	sselid 3979	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑚 ∈ ℝ^*)
118		simplrr 776	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑛 ∈ (𝑍(,)𝑊))
119	12, 118	sselid 3979	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊))) ∧ 𝑧 ∈ (𝑋[,]𝑌)) → 𝑛 ∈ ℝ^*)
120		elioo2 13361	. . . . . . . . . 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 584	. . . . . . . . 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 281	. . . . . . . 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 3175	. . . . . . 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 6899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧)
125	124	nfel1 2919	. . . . . . . . . . 11 ⊢ Ⅎ𝑥((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑚(,)𝑛)
126		nfv 1917	. . . . . . . . . . 11 ⊢ Ⅎ𝑧((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) ∈ (𝑚(,)𝑛)
127		fveq2 6888	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) = ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥))
128	127	eleq1d 2818	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑚(,)𝑛) ↔ ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) ∈ (𝑚(,)𝑛)))
129	125, 126, 128	cbvralw 3303	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑚(,)𝑛) ↔ ∀𝑥 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) ∈ (𝑚(,)𝑛))
130		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋[,]𝑌)) → 𝑥 ∈ (𝑋[,]𝑌))
131	14	fvmptelcdm 7109	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋[,]𝑌)) → 𝐴 ∈ (𝑍(,)𝑊))
132		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) = (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)
133	132	fvmpt2 7006	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝑋[,]𝑌) ∧ 𝐴 ∈ (𝑍(,)𝑊)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) = 𝐴)
134	130, 131, 133	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋[,]𝑌)) → ((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) = 𝐴)
135	134	eleq1d 2818	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋[,]𝑌)) → (((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) ∈ (𝑚(,)𝑛) ↔ 𝐴 ∈ (𝑚(,)𝑛)))
136	135	ralbidva 3175	. . . . . . . . . 10 ⊢ (𝜑 → (∀𝑥 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑥) ∈ (𝑚(,)𝑛) ↔ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛)))
137	129, 136	bitrid 282	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑧 ∈ (𝑋[,]𝑌)((𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)‘𝑧) ∈ (𝑚(,)𝑛) ↔ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛)))
138	137	adantr 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)))
139	1	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑋 ∈ ℝ)
140	2	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑌 ∈ ℝ)
141	3	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑋 ≤ 𝑌)
142	28	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑍 ∈ ℝ^*)
143	19	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑊 ∈ ℝ^*)
144		nfcv 2903	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦𝐴
145		nfcsb1v 3917	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴
146		csbeq1a 3906	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴)
147	144, 145, 146	cbvmpt 5258	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) = (𝑦 ∈ (𝑋[,]𝑌) ↦ ⦋𝑦 / 𝑥⦌𝐴)
148	147, 8	eqeltrrid 2838	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑦 ∈ (𝑋[,]𝑌) ↦ ⦋𝑦 / 𝑥⦌𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝑍(,)𝑊)))
149	148	adantr 481	. . . . . . . . . . 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 2903	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦𝐵
151		nfcsb1v 3917	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
152		csbeq1a 3906	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
153	150, 151, 152	cbvmpt 5258	. . . . . . . . . . . . 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 2838	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑦 ∈ (𝑋(,)𝑌) ↦ ⦋𝑦 / 𝑥⦌𝐵) ∈ (((𝑋(,)𝑌)–cn→ℂ) ∩ 𝐿¹))
156	155	adantr 481	. . . . . . . . . . 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 2903	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑣𝐶
158		nfcsb1v 3917	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑢⦋𝑣 / 𝑢⦌𝐶
159		csbeq1a 3906	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑣 → 𝐶 = ⦋𝑣 / 𝑢⦌𝐶)
160	157, 158, 159	cbvmpt 5258	. . . . . . . . . . . . 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 2838	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑣 ∈ (𝑍(,)𝑊) ↦ ⦋𝑣 / 𝑢⦌𝐶) ∈ ((𝑍(,)𝑊)–cn→ℂ))
163	162	adantr 481	. . . . . . . . . . 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 7416	. . . . . . . . . . . . 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 481	. . . . . . . . . . 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 3895	. . . . . . . . . . 11 ⊢ (𝑣 = ⦋𝑦 / 𝑥⦌𝐴 → ⦋𝑣 / 𝑢⦌𝐶 = ⦋⦋𝑦 / 𝑥⦌𝐴 / 𝑢⦌𝐶)
169		csbeq1 3895	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑋 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑋 / 𝑥⦌𝐴)
170		csbeq1 3895	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑌 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑌 / 𝑥⦌𝐴)
171		simprll 777	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑚 ∈ (𝑍(,)𝑊))
172		simprlr 778	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → 𝑛 ∈ (𝑍(,)𝑊))
173		simprr 771	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑚 ∈ (𝑍(,)𝑊) ∧ 𝑛 ∈ (𝑍(,)𝑊)) ∧ ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))) → ∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛))
174	145	nfel1 2919	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 ∈ (𝑚(,)𝑛)
175	146	eleq1d 2818	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ (𝑚(,)𝑛) ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ (𝑚(,)𝑛)))
176	174, 175	rspc 3600	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝑋[,]𝑌) → (∀𝑥 ∈ (𝑋[,]𝑌)𝐴 ∈ (𝑚(,)𝑛) → ⦋𝑦 / 𝑥⦌𝐴 ∈ (𝑚(,)𝑛)))
177	173, 176	mpan9 507	. . . . . . . . . . 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 25556	. . . . . . . . . 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 25362	. . . . . . . . . . . 12 ⊢ ⨜[⦋𝑋 / 𝑥⦌𝐴 → ⦋𝑌 / 𝑥⦌𝐴]𝐶 d𝑢 = ⨜[⦋𝑋 / 𝑥⦌𝐴 → ⦋𝑌 / 𝑥⦌𝐴]⦋𝑣 / 𝑢⦌𝐶 d𝑣
180		nfcvd 2904	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ ℝ → Ⅎ𝑥𝐾)
181		itgsubst.k	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑋 → 𝐴 = 𝐾)
182	180, 181	csbiegf 3926	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℝ → ⦋𝑋 / 𝑥⦌𝐴 = 𝐾)
183		ditgeq1 25356	. . . . . . . . . . . . . 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 2904	. . . . . . . . . . . . . . 15 ⊢ (𝑌 ∈ ℝ → Ⅎ𝑥𝐿)
186		itgsubst.l	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑌 → 𝐴 = 𝐿)
187	185, 186	csbiegf 3926	. . . . . . . . . . . . . 14 ⊢ (𝑌 ∈ ℝ → ⦋𝑌 / 𝑥⦌𝐴 = 𝐿)
188		ditgeq2 25357	. . . . . . . . . . . . . 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 481	. . . . . . . . . 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 3896	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ⦋𝐴 / 𝑢⦌𝐶 = ⦋⦋𝑦 / 𝑥⦌𝐴 / 𝑢⦌𝐶)
194	193, 152	oveq12d 7423	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (⦋𝐴 / 𝑢⦌𝐶 · 𝐵) = (⦋⦋𝑦 / 𝑥⦌𝐴 / 𝑢⦌𝐶 · ⦋𝑦 / 𝑥⦌𝐵))
195		nfcv 2903	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦(⦋𝐴 / 𝑢⦌𝐶 · 𝐵)
196		nfcv 2903	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥𝐶
197	145, 196	nfcsbw 3919	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⦋⦋𝑦 / 𝑥⦌𝐴 / 𝑢⦌𝐶
198		nfcv 2903	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥 ·
199	197, 198, 151	nfov 7435	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(⦋⦋𝑦 / 𝑥⦌𝐴 / 𝑢⦌𝐶 · ⦋𝑦 / 𝑥⦌𝐵)
200	194, 195, 199	cbvditg 25362	. . . . . . . . . . . 12 ⊢ ⨜[𝑋 → 𝑌](⦋𝐴 / 𝑢⦌𝐶 · 𝐵) d𝑥 = ⨜[𝑋 → 𝑌](⦋⦋𝑦 / 𝑥⦌𝐴 / 𝑢⦌𝐶 · ⦋𝑦 / 𝑥⦌𝐵) d𝑦
201		ioossicc 13406	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋(,)𝑌) ⊆ (𝑋[,]𝑌)
202	201	sseli 3977	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝑋(,)𝑌) → 𝑥 ∈ (𝑋[,]𝑌))
203	202, 131	sylan2 593	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐴 ∈ (𝑍(,)𝑊))
204		nfcvd 2904	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ (𝑍(,)𝑊) → Ⅎ𝑢𝐸)
205		itgsubst.e	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝐴 → 𝐶 = 𝐸)
206	204, 205	csbiegf 3926	. . . . . . . . . . . . . . . 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 7420	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → (⦋𝐴 / 𝑢⦌𝐶 · 𝐵) = (𝐸 · 𝐵))
209	208	itgeq2dv 25290	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∫(𝑋(,)𝑌)(⦋𝐴 / 𝑢⦌𝐶 · 𝐵) d𝑥 = ∫(𝑋(,)𝑌)(𝐸 · 𝐵) d𝑥)
210	3	ditgpos 25364	. . . . . . . . . . . . 13 ⊢ (𝜑 → ⨜[𝑋 → 𝑌](⦋𝐴 / 𝑢⦌𝐶 · 𝐵) d𝑥 = ∫(𝑋(,)𝑌)(⦋𝐴 / 𝑢⦌𝐶 · 𝐵) d𝑥)
211	3	ditgpos 25364	. . . . . . . . . . . . 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 481	. . . . . . . . . 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 457	. . . . . . . 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 239	. . . . . . 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 239	. . . . . 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 242	. . . . 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 3211	. . . 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 241	. . 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 608	. 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 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 ⦋csb 3892 ∩ cin 3946 ⊆ wss 3947 class class class wbr 5147 ↦ cmpt 5230 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 ℝcr 11105 · cmul 11111 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 ℚcq 12928 (,)cioo 13320 [,]cicc 13323 –cn→ccncf 24383 𝐿¹cibl 25125 ∫citg 25126 ⨜cdit 25354 D cdv 25371

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cc 10426 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-symdif 4241 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-omul 8467 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-acn 9933 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-cmp 22882 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-ovol 24972 df-vol 24973 df-mbf 25127 df-itg1 25128 df-itg2 25129 df-ibl 25130 df-itg 25131 df-0p 25178 df-ditg 25355 df-limc 25374 df-dv 25375

This theorem is referenced by: itgsubsticclem 44677

W3C validator