Theorem mtestbdd 26448

Description: Given the hypotheses of the Weierstrass M-test, the convergent function of the sequence is uniformly bounded. (Contributed by Mario Carneiro, 9-Jul-2017.)

Hypotheses

Ref	Expression
mtest.z	⊢ 𝑍 = (ℤ≥‘𝑁)
mtest.n	⊢ (𝜑 → 𝑁 ∈ ℤ)
mtest.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
mtest.f	⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆))
mtest.m	⊢ (𝜑 → 𝑀 ∈ 𝑊)
mtest.c	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑀‘𝑘) ∈ ℝ)
mtest.l	⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → (abs‘((𝐹‘𝑘)‘𝑧)) ≤ (𝑀‘𝑘))
mtest.d	⊢ (𝜑 → seq𝑁( + , 𝑀) ∈ dom ⇝ )
mtest.t	⊢ (𝜑 → seq𝑁( ∘f + , 𝐹)(⇝𝑢‘𝑆)𝑇)

Assertion

Ref	Expression
mtestbdd	⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 (abs‘(𝑇‘𝑧)) ≤ 𝑥)

Distinct variable groups:   𝑥,𝑘,𝑧,𝐹   𝑘,𝑀,𝑥,𝑧   𝑘,𝑁,𝑥,𝑧   𝜑,𝑘,𝑥,𝑧   𝑥,𝑇,𝑧   𝑘,𝑍,𝑥,𝑧   𝑆,𝑘,𝑥,𝑧

Allowed substitution hints:   𝑇(𝑘)   𝑉(𝑥,𝑧,𝑘)   𝑊(𝑥,𝑧,𝑘)

Proof of Theorem mtestbdd

Dummy variables 𝑗 𝑛 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mtest.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
2		mtest.d	. . 3 ⊢ (𝜑 → seq𝑁( + , 𝑀) ∈ dom ⇝ )
3		mtest.z	. . . . . 6 ⊢ 𝑍 = (ℤ≥‘𝑁)
4		mtest.c	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑀‘𝑘) ∈ ℝ)
5	4	recnd 11289	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑀‘𝑘) ∈ ℂ)
6	3, 1, 5	serf 14071	. . . . 5 ⊢ (𝜑 → seq𝑁( + , 𝑀):𝑍⟶ℂ)
7	6	ffvelcdmda 7104	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (seq𝑁( + , 𝑀)‘𝑚) ∈ ℂ)
8	7	ralrimiva 3146	. . 3 ⊢ (𝜑 → ∀𝑚 ∈ 𝑍 (seq𝑁( + , 𝑀)‘𝑚) ∈ ℂ)
9	3	climbdd 15708	. . 3 ⊢ ((𝑁 ∈ ℤ ∧ seq𝑁( + , 𝑀) ∈ dom ⇝ ∧ ∀𝑚 ∈ 𝑍 (seq𝑁( + , 𝑀)‘𝑚) ∈ ℂ) → ∃𝑦 ∈ ℝ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)
10	1, 2, 8, 9	syl3anc 1373	. 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)
11	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → 𝑁 ∈ ℤ)
12		seqfn 14054	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → seq𝑁( ∘f + , 𝐹) Fn (ℤ≥‘𝑁))
13	1, 12	syl 17	. . . . . 6 ⊢ (𝜑 → seq𝑁( ∘f + , 𝐹) Fn (ℤ≥‘𝑁))
14	3	fneq2i 6666	. . . . . 6 ⊢ (seq𝑁( ∘f + , 𝐹) Fn 𝑍 ↔ seq𝑁( ∘f + , 𝐹) Fn (ℤ≥‘𝑁))
15	13, 14	sylibr 234	. . . . 5 ⊢ (𝜑 → seq𝑁( ∘f + , 𝐹) Fn 𝑍)
16		mtest.t	. . . . 5 ⊢ (𝜑 → seq𝑁( ∘f + , 𝐹)(⇝𝑢‘𝑆)𝑇)
17		ulmf2 26427	. . . . 5 ⊢ ((seq𝑁( ∘f + , 𝐹) Fn 𝑍 ∧ seq𝑁( ∘f + , 𝐹)(⇝𝑢‘𝑆)𝑇) → seq𝑁( ∘f + , 𝐹):𝑍⟶(ℂ ↑m 𝑆))
18	15, 16, 17	syl2anc 584	. . . 4 ⊢ (𝜑 → seq𝑁( ∘f + , 𝐹):𝑍⟶(ℂ ↑m 𝑆))
19	18	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → seq𝑁( ∘f + , 𝐹):𝑍⟶(ℂ ↑m 𝑆))
20		simplrl 777	. . . 4 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) → 𝑦 ∈ ℝ)
21		fveq2 6906	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑗)‘𝑥) = ((𝐹‘𝑗)‘𝑧))
22	21	mpteq2dv 5244	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥)) = (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧)))
23	22	seqeq3d 14050	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥))) = seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧))))
24	23	fveq1d 6908	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥)))‘𝑛) = (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧)))‘𝑛))
25		eqid 2737	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑆 ↦ (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥)))‘𝑛)) = (𝑥 ∈ 𝑆 ↦ (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥)))‘𝑛))
26		fvex 6919	. . . . . . . . . 10 ⊢ (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧)))‘𝑛) ∈ V
27	24, 25, 26	fvmpt 7016	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑆 → ((𝑥 ∈ 𝑆 ↦ (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥)))‘𝑛))‘𝑧) = (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧)))‘𝑛))
28	27	adantl 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → ((𝑥 ∈ 𝑆 ↦ (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥)))‘𝑛))‘𝑧) = (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧)))‘𝑛))
29		mtest.f	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆))
30	29	ad3antrrr 730	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → 𝐹:𝑍⟶(ℂ ↑m 𝑆))
31	30	feqmptd 6977	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → 𝐹 = (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))
32	30	ffvelcdmda 7104	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ (ℂ ↑m 𝑆))
33		elmapi 8889	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑗) ∈ (ℂ ↑m 𝑆) → (𝐹‘𝑗):𝑆⟶ℂ)
34	32, 33	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗):𝑆⟶ℂ)
35	34	feqmptd 6977	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑗)‘𝑥)))
36	35	mpteq2dva 5242	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)) = (𝑗 ∈ 𝑍 ↦ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑗)‘𝑥))))
37	31, 36	eqtrd 2777	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → 𝐹 = (𝑗 ∈ 𝑍 ↦ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑗)‘𝑥))))
38	37	seqeq3d 14050	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → seq𝑁( ∘f + , 𝐹) = seq𝑁( ∘f + , (𝑗 ∈ 𝑍 ↦ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑗)‘𝑥)))))
39	38	fveq1d 6908	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → (seq𝑁( ∘f + , 𝐹)‘𝑛) = (seq𝑁( ∘f + , (𝑗 ∈ 𝑍 ↦ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑗)‘𝑥))))‘𝑛))
40		mtest.s	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
41	40	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → 𝑆 ∈ 𝑉)
42		simplr 769	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → 𝑛 ∈ 𝑍)
43	42, 3	eleqtrdi 2851	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → 𝑛 ∈ (ℤ≥‘𝑁))
44		elfzuz 13560	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (𝑁...𝑛) → 𝑘 ∈ (ℤ≥‘𝑁))
45	44, 3	eleqtrrdi 2852	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑁...𝑛) → 𝑘 ∈ 𝑍)
46	45	ssriv 3987	. . . . . . . . . . . 12 ⊢ (𝑁...𝑛) ⊆ 𝑍
47	46	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → (𝑁...𝑛) ⊆ 𝑍)
48	34	ffvelcdmda 7104	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 ∈ 𝑆) → ((𝐹‘𝑗)‘𝑥) ∈ ℂ)
49	48	anasss 466	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ (𝑗 ∈ 𝑍 ∧ 𝑥 ∈ 𝑆)) → ((𝐹‘𝑗)‘𝑥) ∈ ℂ)
50	41, 43, 47, 49	seqof2 14101	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → (seq𝑁( ∘f + , (𝑗 ∈ 𝑍 ↦ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑗)‘𝑥))))‘𝑛) = (𝑥 ∈ 𝑆 ↦ (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥)))‘𝑛)))
51	39, 50	eqtrd 2777	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → (seq𝑁( ∘f + , 𝐹)‘𝑛) = (𝑥 ∈ 𝑆 ↦ (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥)))‘𝑛)))
52	51	fveq1d 6908	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → ((seq𝑁( ∘f + , 𝐹)‘𝑛)‘𝑧) = ((𝑥 ∈ 𝑆 ↦ (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥)))‘𝑛))‘𝑧))
53	45	adantl 481	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → 𝑘 ∈ 𝑍)
54		fveq2 6906	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘))
55	54	fveq1d 6908	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ((𝐹‘𝑗)‘𝑧) = ((𝐹‘𝑘)‘𝑧))
56		eqid 2737	. . . . . . . . . . 11 ⊢ (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧)) = (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧))
57		fvex 6919	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑘)‘𝑧) ∈ V
58	55, 56, 57	fvmpt 7016	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝑍 → ((𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧))‘𝑘) = ((𝐹‘𝑘)‘𝑧))
59	53, 58	syl 17	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → ((𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧))‘𝑘) = ((𝐹‘𝑘)‘𝑧))
60		simplr 769	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑗 ∈ 𝑍) → 𝑧 ∈ 𝑆)
61	34, 60	ffvelcdmd 7105	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑗 ∈ 𝑍) → ((𝐹‘𝑗)‘𝑧) ∈ ℂ)
62	61	fmpttd 7135	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧)):𝑍⟶ℂ)
63	62	ffvelcdmda 7104	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ 𝑍) → ((𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧))‘𝑘) ∈ ℂ)
64	45, 63	sylan2 593	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → ((𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧))‘𝑘) ∈ ℂ)
65	59, 64	eqeltrrd 2842	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → ((𝐹‘𝑘)‘𝑧) ∈ ℂ)
66	59, 43, 65	fsumser 15766	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → Σ𝑘 ∈ (𝑁...𝑛)((𝐹‘𝑘)‘𝑧) = (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧)))‘𝑛))
67	28, 52, 66	3eqtr4d 2787	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → ((seq𝑁( ∘f + , 𝐹)‘𝑛)‘𝑧) = Σ𝑘 ∈ (𝑁...𝑛)((𝐹‘𝑘)‘𝑧))
68	67	fveq2d 6910	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → (abs‘((seq𝑁( ∘f + , 𝐹)‘𝑛)‘𝑧)) = (abs‘Σ𝑘 ∈ (𝑁...𝑛)((𝐹‘𝑘)‘𝑧)))
69		fzfid 14014	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → (𝑁...𝑛) ∈ Fin)
70	69, 65	fsumcl 15769	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → Σ𝑘 ∈ (𝑁...𝑛)((𝐹‘𝑘)‘𝑧) ∈ ℂ)
71	70	abscld 15475	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)((𝐹‘𝑘)‘𝑧)) ∈ ℝ)
72	65	abscld 15475	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → (abs‘((𝐹‘𝑘)‘𝑧)) ∈ ℝ)
73	69, 72	fsumrecl 15770	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(abs‘((𝐹‘𝑘)‘𝑧)) ∈ ℝ)
74	20	adantr 480	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → 𝑦 ∈ ℝ)
75	69, 65	fsumabs 15837	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)((𝐹‘𝑘)‘𝑧)) ≤ Σ𝑘 ∈ (𝑁...𝑛)(abs‘((𝐹‘𝑘)‘𝑧)))
76		simp-4l 783	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → 𝜑)
77	76, 53, 4	syl2anc 584	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → (𝑀‘𝑘) ∈ ℝ)
78	69, 77	fsumrecl 15770	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀‘𝑘) ∈ ℝ)
79		simplr 769	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → 𝑧 ∈ 𝑆)
80		mtest.l	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → (abs‘((𝐹‘𝑘)‘𝑧)) ≤ (𝑀‘𝑘))
81	76, 53, 79, 80	syl12anc 837	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → (abs‘((𝐹‘𝑘)‘𝑧)) ≤ (𝑀‘𝑘))
82	69, 72, 77, 81	fsumle 15835	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(abs‘((𝐹‘𝑘)‘𝑧)) ≤ Σ𝑘 ∈ (𝑁...𝑛)(𝑀‘𝑘))
83	78	recnd 11289	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀‘𝑘) ∈ ℂ)
84	83	abscld 15475	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)(𝑀‘𝑘)) ∈ ℝ)
85	78	leabsd 15453	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀‘𝑘) ≤ (abs‘Σ𝑘 ∈ (𝑁...𝑛)(𝑀‘𝑘)))
86		eqidd 2738	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → (𝑀‘𝑘) = (𝑀‘𝑘))
87	76, 53, 5	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → (𝑀‘𝑘) ∈ ℂ)
88	86, 43, 87	fsumser 15766	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀‘𝑘) = (seq𝑁( + , 𝑀)‘𝑛))
89	88	fveq2d 6910	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)(𝑀‘𝑘)) = (abs‘(seq𝑁( + , 𝑀)‘𝑛)))
90		simprr 773	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)
91		fveq2 6906	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (seq𝑁( + , 𝑀)‘𝑚) = (seq𝑁( + , 𝑀)‘𝑛))
92	91	fveq2d 6910	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (abs‘(seq𝑁( + , 𝑀)‘𝑚)) = (abs‘(seq𝑁( + , 𝑀)‘𝑛)))
93	92	breq1d 5153	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → ((abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦 ↔ (abs‘(seq𝑁( + , 𝑀)‘𝑛)) ≤ 𝑦))
94	93	rspccva 3621	. . . . . . . . . . . 12 ⊢ ((∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦 ∧ 𝑛 ∈ 𝑍) → (abs‘(seq𝑁( + , 𝑀)‘𝑛)) ≤ 𝑦)
95	90, 94	sylan 580	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) → (abs‘(seq𝑁( + , 𝑀)‘𝑛)) ≤ 𝑦)
96	95	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → (abs‘(seq𝑁( + , 𝑀)‘𝑛)) ≤ 𝑦)
97	89, 96	eqbrtrd 5165	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)(𝑀‘𝑘)) ≤ 𝑦)
98	78, 84, 74, 85, 97	letrd 11418	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀‘𝑘) ≤ 𝑦)
99	73, 78, 74, 82, 98	letrd 11418	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(abs‘((𝐹‘𝑘)‘𝑧)) ≤ 𝑦)
100	71, 73, 74, 75, 99	letrd 11418	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)((𝐹‘𝑘)‘𝑧)) ≤ 𝑦)
101	68, 100	eqbrtrd 5165	. . . . 5 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → (abs‘((seq𝑁( ∘f + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑦)
102	101	ralrimiva 3146	. . . 4 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) → ∀𝑧 ∈ 𝑆 (abs‘((seq𝑁( ∘f + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑦)
103		brralrspcev 5203	. . . 4 ⊢ ((𝑦 ∈ ℝ ∧ ∀𝑧 ∈ 𝑆 (abs‘((seq𝑁( ∘f + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 (abs‘((seq𝑁( ∘f + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑥)
104	20, 102, 103	syl2anc 584	. . 3 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 (abs‘((seq𝑁( ∘f + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑥)
105	16	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → seq𝑁( ∘f + , 𝐹)(⇝𝑢‘𝑆)𝑇)
106	3, 11, 19, 104, 105	ulmbdd 26441	. 2 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 (abs‘(𝑇‘𝑧)) ≤ 𝑥)
107	10, 106	rexlimddv 3161	1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 (abs‘(𝑇‘𝑧)) ≤ 𝑥)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   ⊆ wss 3951   class class class wbr 5143   ↦ cmpt 5225  dom cdm 5685   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ∘_f cof 7695   ↑_m cmap 8866  ℂcc 11153  ℝcr 11154   + caddc 11158   ≤ cle 11296  ℤcz 12613  ℤ_≥cuz 12878  ...cfz 13547  seqcseq 14042  abscabs 15273   ⇝ cli 15520  Σcsu 15722  ⇝_𝑢culm 26419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-ico 13393  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-ulm 26420

This theorem is referenced by:  lgamgulmlem6  27077