Theorem mtestbdd 26732

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 11337	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑀‘𝑘) ∈ ℂ)
6	3, 1, 5	serf 14173	. . . . 5 ⊢ (𝜑 → seq𝑁( + , 𝑀):𝑍⟶ℂ)
7	6	ffvelcdmda 7084	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (seq𝑁( + , 𝑀)‘𝑚) ∈ ℂ)
8	7	ralrimiva 3155	. . 3 ⊢ (𝜑 → ∀𝑚 ∈ 𝑍 (seq𝑁( + , 𝑀)‘𝑚) ∈ ℂ)
9	3	climbdd 15839	. . 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 1398	. 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)
11	1	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → 𝑁 ∈ ℤ)
12		seqfn 14156	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → seq𝑁( ∘f + , 𝐹) Fn (ℤ≥‘𝑁))
13	1, 12	syl 18	. . . . . 6 ⊢ (𝜑 → seq𝑁( ∘f + , 𝐹) Fn (ℤ≥‘𝑁))
14	3	fneq2i 6637	. . . . . 6 ⊢ (seq𝑁( ∘f + , 𝐹) Fn 𝑍 ↔ seq𝑁( ∘f + , 𝐹) Fn (ℤ≥‘𝑁))
15	13, 14	sylibr 237	. . . . 5 ⊢ (𝜑 → seq𝑁( ∘f + , 𝐹) Fn 𝑍)
16		mtest.t	. . . . 5 ⊢ (𝜑 → seq𝑁( ∘f + , 𝐹)(⇝𝑢‘𝑆)𝑇)
17		ulmf2 26711	. . . . 5 ⊢ ((seq𝑁( ∘f + , 𝐹) Fn 𝑍 ∧ seq𝑁( ∘f + , 𝐹)(⇝𝑢‘𝑆)𝑇) → seq𝑁( ∘f + , 𝐹):𝑍⟶(ℂ ↑m 𝑆))
18	15, 16, 17	syl2anc 596	. . . 4 ⊢ (𝜑 → seq𝑁( ∘f + , 𝐹):𝑍⟶(ℂ ↑m 𝑆))
19	18	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → seq𝑁( ∘f + , 𝐹):𝑍⟶(ℂ ↑m 𝑆))
20		simplrl 789	. . . 4 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) → 𝑦 ∈ ℝ)
21		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑗)‘𝑥) = ((𝐹‘𝑗)‘𝑧))
22	21	mpteq2dv 5199	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥)) = (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧)))
23	22	seqeq3d 14152	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥))) = seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧))))
24	23	fveq1d 6887	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥)))‘𝑛) = (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧)))‘𝑛))
25		eqid 2761	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑆 ↦ (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥)))‘𝑛)) = (𝑥 ∈ 𝑆 ↦ (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥)))‘𝑛))
26		fvex 6898	. . . . . . . . . 10 ⊢ (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧)))‘𝑛) ∈ V
27	24, 25, 26	fvmpt 6993	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑆 → ((𝑥 ∈ 𝑆 ↦ (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥)))‘𝑛))‘𝑧) = (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧)))‘𝑛))
28	27	adantl 487	. . . . . . . 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 743	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → 𝐹:𝑍⟶(ℂ ↑m 𝑆))
31	30	feqmptd 6953	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → 𝐹 = (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))
32	30	ffvelcdmda 7084	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ (ℂ ↑m 𝑆))
33		elmapi 8869	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑗) ∈ (ℂ ↑m 𝑆) → (𝐹‘𝑗):𝑆⟶ℂ)
34	32, 33	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗):𝑆⟶ℂ)
35	34	feqmptd 6953	. . . . . . . . . . . . . 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 5198	. . . . . . . . . . . . 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 2796	. . . . . . . . . . . 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 14152	. . . . . . . . . . 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 6887	. . . . . . . . . 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 743	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → 𝑆 ∈ 𝑉)
42		simplr 781	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → 𝑛 ∈ 𝑍)
43	42, 3	eleqtrdi 2871	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → 𝑛 ∈ (ℤ≥‘𝑁))
44		elfzuz 13652	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (𝑁...𝑛) → 𝑘 ∈ (ℤ≥‘𝑁))
45	44, 3	eleqtrrdi 2872	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑁...𝑛) → 𝑘 ∈ 𝑍)
46	45	ssriv 3935	. . . . . . . . . . . 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 7084	. . . . . . . . . . . 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 472	. . . . . . . . . . 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 14203	. . . . . . . . . 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 2796	. . . . . . . . 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 6887	. . . . . . . 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 487	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → 𝑘 ∈ 𝑍)
54		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘))
55	54	fveq1d 6887	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ((𝐹‘𝑗)‘𝑧) = ((𝐹‘𝑘)‘𝑧))
56		eqid 2761	. . . . . . . . . . 11 ⊢ (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧)) = (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧))
57		fvex 6898	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑘)‘𝑧) ∈ V
58	55, 56, 57	fvmpt 6993	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝑍 → ((𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧))‘𝑘) = ((𝐹‘𝑘)‘𝑧))
59	53, 58	syl 18	. . . . . . . . 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 781	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑗 ∈ 𝑍) → 𝑧 ∈ 𝑆)
61	34, 60	ffvelcdmd 7085	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑗 ∈ 𝑍) → ((𝐹‘𝑗)‘𝑧) ∈ ℂ)
62	61	fmpttd 7115	. . . . . . . . . . . 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 7084	. . . . . . . . . . 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 605	. . . . . . . . . 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 2862	. . . . . . . . 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 15896	. . . . . . . 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 2806	. . . . . . 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 6889	. . . . . 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 14116	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → (𝑁...𝑛) ∈ Fin)
70	69, 65	fsumcl 15899	. . . . . . . 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 15606	. . . . . . 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 15606	. . . . . . . 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 15900	. . . . . . 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 486	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → 𝑦 ∈ ℝ)
75	69, 65	fsumabs 15968	. . . . . . 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 795	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → 𝜑)
77	76, 53, 4	syl2anc 596	. . . . . . . . 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 15900	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀‘𝑘) ∈ ℝ)
79		simplr 781	. . . . . . . . . 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 850	. . . . . . . . 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 15966	. . . . . . . 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 11337	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀‘𝑘) ∈ ℂ)
84	83	abscld 15606	. . . . . . . . 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 15582	. . . . . . . . 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 2762	. . . . . . . . . . . 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 596	. . . . . . . . . . . 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 15896	. . . . . . . . . . 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 6889	. . . . . . . . . 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 785	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)
91		fveq2 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (seq𝑁( + , 𝑀)‘𝑚) = (seq𝑁( + , 𝑀)‘𝑛))
92	91	fveq2d 6889	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (abs‘(seq𝑁( + , 𝑀)‘𝑚)) = (abs‘(seq𝑁( + , 𝑀)‘𝑛)))
93	92	breq1d 5113	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → ((abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦 ↔ (abs‘(seq𝑁( + , 𝑀)‘𝑛)) ≤ 𝑦))
94	93	rspccva 3576	. . . . . . . . . . . 12 ⊢ ((∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦 ∧ 𝑛 ∈ 𝑍) → (abs‘(seq𝑁( + , 𝑀)‘𝑛)) ≤ 𝑦)
95	90, 94	sylan 592	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) → (abs‘(seq𝑁( + , 𝑀)‘𝑛)) ≤ 𝑦)
96	95	adantr 486	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → (abs‘(seq𝑁( + , 𝑀)‘𝑛)) ≤ 𝑦)
97	89, 96	eqbrtrd 5127	. . . . . . . . 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 11467	. . . . . . . 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 11467	. . . . . . 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 11467	. . . . . 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 5127	. . . . 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 3155	. . . 4 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) → ∀𝑧 ∈ 𝑆 (abs‘((seq𝑁( ∘f + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑦)
103		brralrspcev 5165	. . . 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 596	. . 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 486	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → seq𝑁( ∘f + , 𝐹)(⇝𝑢‘𝑆)𝑇)
106	3, 11, 19, 104, 105	ulmbdd 26725	. 2 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 (abs‘(𝑇‘𝑧)) ≤ 𝑥)
107	10, 106	rexlimddv 3170	1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 (abs‘(𝑇‘𝑧)) ≤ 𝑥)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087   ⊆ wss 3899   class class class wbr 5103   ↦ cmpt 5186  dom cdm 5651   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ∘_f cof 7691   ↑_m cmap 8847  ℂcc 11198  ℝcr 11199   + caddc 11203   ≤ cle 11344  ℤcz 12693  ℤ_≥cuz 12965  ...cfz 13639  seqcseq 14144  abscabs 15401   ⇝ cli 15651  Σcsu 15853  ⇝_𝑢culm 26703

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-inf2 9642  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-map 8849  df-pm 8850  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-sup 9434  df-oi 9504  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-3 12406  df-n0 12607  df-z 12694  df-uz 12966  df-rp 13121  df-ico 13482  df-fz 13640  df-fzo 13789  df-seq 14145  df-exp 14205  df-hash 14475  df-cj 15266  df-re 15267  df-im 15268  df-sqrt 15402  df-abs 15403  df-clim 15655  df-sum 15854  df-ulm 26704

This theorem is used by:  lgamgulmlem6  27361