Theorem mtestbdd 26314

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 11202	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑀‘𝑘) ∈ ℂ)
6	3, 1, 5	serf 13995	. . . . 5 ⊢ (𝜑 → seq𝑁( + , 𝑀):𝑍⟶ℂ)
7	6	ffvelcdmda 7056	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (seq𝑁( + , 𝑀)‘𝑚) ∈ ℂ)
8	7	ralrimiva 3125	. . 3 ⊢ (𝜑 → ∀𝑚 ∈ 𝑍 (seq𝑁( + , 𝑀)‘𝑚) ∈ ℂ)
9	3	climbdd 15638	. . 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 13978	. . . . . . 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 6616	. . . . . 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 26293	. . . . 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 776	. . . 4 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) → 𝑦 ∈ ℝ)
21		fveq2 6858	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑗)‘𝑥) = ((𝐹‘𝑗)‘𝑧))
22	21	mpteq2dv 5201	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥)) = (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧)))
23	22	seqeq3d 13974	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥))) = seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧))))
24	23	fveq1d 6860	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥)))‘𝑛) = (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧)))‘𝑛))
25		eqid 2729	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑆 ↦ (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥)))‘𝑛)) = (𝑥 ∈ 𝑆 ↦ (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥)))‘𝑛))
26		fvex 6871	. . . . . . . . . 10 ⊢ (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧)))‘𝑛) ∈ V
27	24, 25, 26	fvmpt 6968	. . . . . . . . 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 6929	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → 𝐹 = (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))
32	30	ffvelcdmda 7056	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ (ℂ ↑m 𝑆))
33		elmapi 8822	. . . . . . . . . . . . . . . 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 6929	. . . . . . . . . . . . . 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 5200	. . . . . . . . . . . . 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 2764	. . . . . . . . . . . 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 13974	. . . . . . . . . . 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 6860	. . . . . . . . . 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 768	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → 𝑛 ∈ 𝑍)
43	42, 3	eleqtrdi 2838	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → 𝑛 ∈ (ℤ≥‘𝑁))
44		elfzuz 13481	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (𝑁...𝑛) → 𝑘 ∈ (ℤ≥‘𝑁))
45	44, 3	eleqtrrdi 2839	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑁...𝑛) → 𝑘 ∈ 𝑍)
46	45	ssriv 3950	. . . . . . . . . . . 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 7056	. . . . . . . . . . . 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 14025	. . . . . . . . . 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 2764	. . . . . . . . 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 6860	. . . . . . . 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 6858	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘))
55	54	fveq1d 6860	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ((𝐹‘𝑗)‘𝑧) = ((𝐹‘𝑘)‘𝑧))
56		eqid 2729	. . . . . . . . . . 11 ⊢ (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧)) = (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧))
57		fvex 6871	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑘)‘𝑧) ∈ V
58	55, 56, 57	fvmpt 6968	. . . . . . . . . 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 768	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑗 ∈ 𝑍) → 𝑧 ∈ 𝑆)
61	34, 60	ffvelcdmd 7057	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑗 ∈ 𝑍) → ((𝐹‘𝑗)‘𝑧) ∈ ℂ)
62	61	fmpttd 7087	. . . . . . . . . . . 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 7056	. . . . . . . . . . 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 2829	. . . . . . . . 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 15696	. . . . . . . 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 2774	. . . . . . 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 6862	. . . . . 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 13938	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → (𝑁...𝑛) ∈ Fin)
70	69, 65	fsumcl 15699	. . . . . . . 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 15405	. . . . . . 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 15405	. . . . . . . 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 15700	. . . . . . 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 15767	. . . . . . 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 782	. . . . . . . . . 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 15700	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀‘𝑘) ∈ ℝ)
79		simplr 768	. . . . . . . . . 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 836	. . . . . . . . 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 15765	. . . . . . . 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 11202	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀‘𝑘) ∈ ℂ)
84	83	abscld 15405	. . . . . . . . 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 15381	. . . . . . . . 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 2730	. . . . . . . . . . . 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 15696	. . . . . . . . . . 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 6862	. . . . . . . . . 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 772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)
91		fveq2 6858	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (seq𝑁( + , 𝑀)‘𝑚) = (seq𝑁( + , 𝑀)‘𝑛))
92	91	fveq2d 6862	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (abs‘(seq𝑁( + , 𝑀)‘𝑚)) = (abs‘(seq𝑁( + , 𝑀)‘𝑛)))
93	92	breq1d 5117	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → ((abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦 ↔ (abs‘(seq𝑁( + , 𝑀)‘𝑛)) ≤ 𝑦))
94	93	rspccva 3587	. . . . . . . . . . . 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 5129	. . . . . . . . 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 11331	. . . . . . . 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 11331	. . . . . . 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 11331	. . . . . 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 5129	. . . . 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 3125	. . . 4 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) → ∀𝑧 ∈ 𝑆 (abs‘((seq𝑁( ∘f + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑦)
103		brralrspcev 5167	. . . 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 26307	. 2 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 (abs‘(𝑇‘𝑧)) ≤ 𝑥)
107	10, 106	rexlimddv 3140	1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 (abs‘(𝑇‘𝑧)) ≤ 𝑥)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ⊆ wss 3914   class class class wbr 5107   ↦ cmpt 5188  dom cdm 5638   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ∘_f cof 7651   ↑_m cmap 8799  ℂcc 11066  ℝcr 11067   + caddc 11071   ≤ cle 11209  ℤcz 12529  ℤ_≥cuz 12793  ...cfz 13468  seqcseq 13966  abscabs 15200   ⇝ cli 15450  Σcsu 15652  ⇝_𝑢culm 26285

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-inf2 9594  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-pm 8802  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-sup 9393  df-oi 9463  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-div 11836  df-nn 12187  df-2 12249  df-3 12250  df-n0 12443  df-z 12530  df-uz 12794  df-rp 12952  df-ico 13312  df-fz 13469  df-fzo 13616  df-seq 13967  df-exp 14027  df-hash 14296  df-cj 15065  df-re 15066  df-im 15067  df-sqrt 15201  df-abs 15202  df-clim 15454  df-sum 15653  df-ulm 26286

This theorem is referenced by:  lgamgulmlem6  26944