Theorem mtestbdd 24676

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 10515	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑀‘𝑘) ∈ ℂ)
6	3, 1, 5	serf 13248	. . . . 5 ⊢ (𝜑 → seq𝑁( + , 𝑀):𝑍⟶ℂ)
7	6	ffvelrnda 6716	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (seq𝑁( + , 𝑀)‘𝑚) ∈ ℂ)
8	7	ralrimiva 3149	. . 3 ⊢ (𝜑 → ∀𝑚 ∈ 𝑍 (seq𝑁( + , 𝑀)‘𝑚) ∈ ℂ)
9	3	climbdd 14862	. . 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 1364	. 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)
11	1	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → 𝑁 ∈ ℤ)
12		seqfn 13231	. . . . . . 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 6321	. . . . . 6 ⊢ (seq𝑁( ∘𝑓 + , 𝐹) Fn 𝑍 ↔ seq𝑁( ∘𝑓 + , 𝐹) Fn (ℤ≥‘𝑁))
15	13, 14	sylibr 235	. . . . 5 ⊢ (𝜑 → seq𝑁( ∘𝑓 + , 𝐹) Fn 𝑍)
16		mtest.t	. . . . 5 ⊢ (𝜑 → seq𝑁( ∘𝑓 + , 𝐹)(⇝𝑢‘𝑆)𝑇)
17		ulmf2 24655	. . . . 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 481	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → seq𝑁( ∘𝑓 + , 𝐹):𝑍⟶(ℂ ↑𝑚 𝑆))
20		simplrl 773	. . . 4 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) → 𝑦 ∈ ℝ)
21		fveq2 6538	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑗)‘𝑥) = ((𝐹‘𝑗)‘𝑧))
22	21	mpteq2dv 5056	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥)) = (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧)))
23	22	seqeq3d 13227	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥))) = seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧))))
24	23	fveq1d 6540	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥)))‘𝑛) = (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧)))‘𝑛))
25		eqid 2795	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑆 ↦ (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥)))‘𝑛)) = (𝑥 ∈ 𝑆 ↦ (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥)))‘𝑛))
26		fvex 6551	. . . . . . . . . 10 ⊢ (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧)))‘𝑛) ∈ V
27	24, 25, 26	fvmpt 6635	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑆 → ((𝑥 ∈ 𝑆 ↦ (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑥)))‘𝑛))‘𝑧) = (seq𝑁( + , (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧)))‘𝑛))
28	27	adantl 482	. . . . . . . 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 726	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → 𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))
31	30	feqmptd 6601	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → 𝐹 = (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)))
32	30	ffvelrnda 6716	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ (ℂ ↑𝑚 𝑆))
33		elmapi 8278	. . . . . . . . . . . . . . . 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 6601	. . . . . . . . . . . . . 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 5055	. . . . . . . . . . . . 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 2831	. . . . . . . . . . . 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 13227	. . . . . . . . . . 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 6540	. . . . . . . . . 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 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → 𝑆 ∈ 𝑉)
42		simplr 765	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → 𝑛 ∈ 𝑍)
43	42, 3	syl6eleq 2893	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → 𝑛 ∈ (ℤ≥‘𝑁))
44		elfzuz 12754	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (𝑁...𝑛) → 𝑘 ∈ (ℤ≥‘𝑁))
45	44, 3	syl6eleqr 2894	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝑁...𝑛) → 𝑘 ∈ 𝑍)
46	45	ssriv 3893	. . . . . . . . . . . 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	ffvelrnda 6716	. . . . . . . . . . . 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 467	. . . . . . . . . . 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 13278	. . . . . . . . . 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 2831	. . . . . . . . 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 6540	. . . . . . . 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 482	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → 𝑘 ∈ 𝑍)
54		fveq2 6538	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘))
55	54	fveq1d 6540	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ((𝐹‘𝑗)‘𝑧) = ((𝐹‘𝑘)‘𝑧))
56		eqid 2795	. . . . . . . . . . 11 ⊢ (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧)) = (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧))
57		fvex 6551	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑘)‘𝑧) ∈ V
58	55, 56, 57	fvmpt 6635	. . . . . . . . . 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 765	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑗 ∈ 𝑍) → 𝑧 ∈ 𝑆)
61	34, 60	ffvelrnd 6717	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) ∧ 𝑗 ∈ 𝑍) → ((𝐹‘𝑗)‘𝑧) ∈ ℂ)
62	61	fmpttd 6742	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑧)):𝑍⟶ℂ)
63	62	ffvelrnda 6716	. . . . . . . . . . 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 592	. . . . . . . . . 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 2884	. . . . . . . . 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 14920	. . . . . . . 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 2841	. . . . . . 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 6542	. . . . . 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 13191	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → (𝑁...𝑛) ∈ Fin)
70	69, 65	fsumcl 14923	. . . . . . . 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 14630	. . . . . . 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 14630	. . . . . . . 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 14924	. . . . . . 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 481	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → 𝑦 ∈ ℝ)
75	69, 65	fsumabs 14989	. . . . . . 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 779	. . . . . . . . . 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 14924	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀‘𝑘) ∈ ℝ)
79		simplr 765	. . . . . . . . . 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 833	. . . . . . . . 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 14987	. . . . . . . 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 10515	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀‘𝑘) ∈ ℂ)
84	83	abscld 14630	. . . . . . . . 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 14608	. . . . . . . . 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 2796	. . . . . . . . . . . 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 14920	. . . . . . . . . . 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 6542	. . . . . . . . . 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 769	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)
91		fveq2 6538	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (seq𝑁( + , 𝑀)‘𝑚) = (seq𝑁( + , 𝑀)‘𝑛))
92	91	fveq2d 6542	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (abs‘(seq𝑁( + , 𝑀)‘𝑚)) = (abs‘(seq𝑁( + , 𝑀)‘𝑛)))
93	92	breq1d 4972	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → ((abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦 ↔ (abs‘(seq𝑁( + , 𝑀)‘𝑛)) ≤ 𝑦))
94	93	rspccva 3558	. . . . . . . . . . . 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 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → (abs‘(seq𝑁( + , 𝑀)‘𝑛)) ≤ 𝑦)
97	89, 96	eqbrtrd 4984	. . . . . . . . 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 10644	. . . . . . . 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 10644	. . . . . . 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 10644	. . . . . 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 4984	. . . . 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 3149	. . . 4 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛 ∈ 𝑍) → ∀𝑧 ∈ 𝑆 (abs‘((seq𝑁( ∘𝑓 + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑦)
103		brralrspcev 5022	. . . 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 481	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → seq𝑁( ∘𝑓 + , 𝐹)(⇝𝑢‘𝑆)𝑇)
106	3, 11, 19, 104, 105	ulmbdd 24669	. 2 ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚 ∈ 𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 (abs‘(𝑇‘𝑧)) ≤ 𝑥)
107	10, 106	rexlimddv 3254	1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 (abs‘(𝑇‘𝑧)) ≤ 𝑥)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1522   ∈ wcel 2081  ∀wral 3105  ∃wrex 3106   ⊆ wss 3859   class class class wbr 4962   ↦ cmpt 5041  dom cdm 5443   Fn wfn 6220  ⟶wf 6221  ‘cfv 6225  (class class class)co 7016   ∘_𝑓 cof 7265   ↑_𝑚 cmap 8256  ℂcc 10381  ℝcr 10382   + caddc 10386   ≤ cle 10522  ℤcz 11829  ℤ_≥cuz 12093  ...cfz 12742  seqcseq 13219  abscabs 14427   ⇝ cli 14675  Σcsu 14876  ⇝_𝑢culm 24647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460  ax-pre-sup 10461

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-of 7267  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-er 8139  df-map 8258  df-pm 8259  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-sup 8752  df-oi 8820  df-card 9214  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-div 11146  df-nn 11487  df-2 11548  df-3 11549  df-n0 11746  df-z 11830  df-uz 12094  df-rp 12240  df-ico 12594  df-fz 12743  df-fzo 12884  df-seq 13220  df-exp 13280  df-hash 13541  df-cj 14292  df-re 14293  df-im 14294  df-sqrt 14428  df-abs 14429  df-clim 14679  df-sum 14877  df-ulm 24648

This theorem is referenced by:  lgamgulmlem6  25293