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Theorem ulmuni 24980

Description: A sequence of functions uniformly converges to at most one limit. (Contributed by Mario Carneiro, 5-Jul-2017.)

**Assertion**
Ref	Expression
ulmuni	⊢ ((𝐹(⇝_𝑢‘𝑆)𝐺 ∧ 𝐹(⇝_𝑢‘𝑆)𝐻) → 𝐺 = 𝐻)

Proof of Theorem ulmuni Dummy variables 𝑖 𝑘 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ulmcl 24969	. . . 4 ⊢ (𝐹(⇝_𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ)
2	1	adantr 483	. . 3 ⊢ ((𝐹(⇝_𝑢‘𝑆)𝐺 ∧ 𝐹(⇝_𝑢‘𝑆)𝐻) → 𝐺:𝑆⟶ℂ)
3	2	ffnd 6515	. 2 ⊢ ((𝐹(⇝_𝑢‘𝑆)𝐺 ∧ 𝐹(⇝_𝑢‘𝑆)𝐻) → 𝐺 Fn 𝑆)
4		ulmcl 24969	. . . 4 ⊢ (𝐹(⇝_𝑢‘𝑆)𝐻 → 𝐻:𝑆⟶ℂ)
5	4	adantl 484	. . 3 ⊢ ((𝐹(⇝_𝑢‘𝑆)𝐺 ∧ 𝐹(⇝_𝑢‘𝑆)𝐻) → 𝐻:𝑆⟶ℂ)
6	5	ffnd 6515	. 2 ⊢ ((𝐹(⇝_𝑢‘𝑆)𝐺 ∧ 𝐹(⇝_𝑢‘𝑆)𝐻) → 𝐻 Fn 𝑆)
7		eqid 2821	. . . . 5 ⊢ (ℤ_≥‘𝑛) = (ℤ_≥‘𝑛)
8		simplr 767	. . . . 5 ⊢ (((((𝐹(⇝_𝑢‘𝑆)𝐺 ∧ 𝐹(⇝_𝑢‘𝑆)𝐻) ∧ 𝑥 ∈ 𝑆) ∧ 𝑛 ∈ ℤ) ∧ 𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆)) → 𝑛 ∈ ℤ)
9		simpr 487	. . . . 5 ⊢ (((((𝐹(⇝_𝑢‘𝑆)𝐺 ∧ 𝐹(⇝_𝑢‘𝑆)𝐻) ∧ 𝑥 ∈ 𝑆) ∧ 𝑛 ∈ ℤ) ∧ 𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆)) → 𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆))
10		simpllr 774	. . . . 5 ⊢ (((((𝐹(⇝_𝑢‘𝑆)𝐺 ∧ 𝐹(⇝_𝑢‘𝑆)𝐻) ∧ 𝑥 ∈ 𝑆) ∧ 𝑛 ∈ ℤ) ∧ 𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆)) → 𝑥 ∈ 𝑆)
11		fvex 6683	. . . . . . 7 ⊢ (ℤ_≥‘𝑛) ∈ V
12	11	mptex 6986	. . . . . 6 ⊢ (𝑖 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑖)‘𝑥)) ∈ V
13	12	a1i 11	. . . . 5 ⊢ (((((𝐹(⇝_𝑢‘𝑆)𝐺 ∧ 𝐹(⇝_𝑢‘𝑆)𝐻) ∧ 𝑥 ∈ 𝑆) ∧ 𝑛 ∈ ℤ) ∧ 𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆)) → (𝑖 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑖)‘𝑥)) ∈ V)
14		fveq2 6670	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝐹‘𝑖) = (𝐹‘𝑘))
15	14	fveq1d 6672	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → ((𝐹‘𝑖)‘𝑥) = ((𝐹‘𝑘)‘𝑥))
16		eqid 2821	. . . . . . . 8 ⊢ (𝑖 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑖)‘𝑥)) = (𝑖 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑖)‘𝑥))
17		fvex 6683	. . . . . . . 8 ⊢ ((𝐹‘𝑘)‘𝑥) ∈ V
18	15, 16, 17	fvmpt 6768	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘𝑛) → ((𝑖 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑖)‘𝑥))‘𝑘) = ((𝐹‘𝑘)‘𝑥))
19	18	eqcomd 2827	. . . . . 6 ⊢ (𝑘 ∈ (ℤ_≥‘𝑛) → ((𝐹‘𝑘)‘𝑥) = ((𝑖 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑖)‘𝑥))‘𝑘))
20	19	adantl 484	. . . . 5 ⊢ ((((((𝐹(⇝_𝑢‘𝑆)𝐺 ∧ 𝐹(⇝_𝑢‘𝑆)𝐻) ∧ 𝑥 ∈ 𝑆) ∧ 𝑛 ∈ ℤ) ∧ 𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆)) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑘)‘𝑥) = ((𝑖 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑖)‘𝑥))‘𝑘))
21		simp-4l 781	. . . . 5 ⊢ (((((𝐹(⇝_𝑢‘𝑆)𝐺 ∧ 𝐹(⇝_𝑢‘𝑆)𝐻) ∧ 𝑥 ∈ 𝑆) ∧ 𝑛 ∈ ℤ) ∧ 𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆)) → 𝐹(⇝_𝑢‘𝑆)𝐺)
22	7, 8, 9, 10, 13, 20, 21	ulmclm 24975	. . . 4 ⊢ (((((𝐹(⇝_𝑢‘𝑆)𝐺 ∧ 𝐹(⇝_𝑢‘𝑆)𝐻) ∧ 𝑥 ∈ 𝑆) ∧ 𝑛 ∈ ℤ) ∧ 𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆)) → (𝑖 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑖)‘𝑥)) ⇝ (𝐺‘𝑥))
23		simp-4r 782	. . . . 5 ⊢ (((((𝐹(⇝_𝑢‘𝑆)𝐺 ∧ 𝐹(⇝_𝑢‘𝑆)𝐻) ∧ 𝑥 ∈ 𝑆) ∧ 𝑛 ∈ ℤ) ∧ 𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆)) → 𝐹(⇝_𝑢‘𝑆)𝐻)
24	7, 8, 9, 10, 13, 20, 23	ulmclm 24975	. . . 4 ⊢ (((((𝐹(⇝_𝑢‘𝑆)𝐺 ∧ 𝐹(⇝_𝑢‘𝑆)𝐻) ∧ 𝑥 ∈ 𝑆) ∧ 𝑛 ∈ ℤ) ∧ 𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆)) → (𝑖 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑖)‘𝑥)) ⇝ (𝐻‘𝑥))
25		climuni 14909	. . . 4 ⊢ (((𝑖 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑖)‘𝑥)) ⇝ (𝐺‘𝑥) ∧ (𝑖 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑖)‘𝑥)) ⇝ (𝐻‘𝑥)) → (𝐺‘𝑥) = (𝐻‘𝑥))
26	22, 24, 25	syl2anc 586	. . 3 ⊢ (((((𝐹(⇝_𝑢‘𝑆)𝐺 ∧ 𝐹(⇝_𝑢‘𝑆)𝐻) ∧ 𝑥 ∈ 𝑆) ∧ 𝑛 ∈ ℤ) ∧ 𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆)) → (𝐺‘𝑥) = (𝐻‘𝑥))
27		ulmf 24970	. . . 4 ⊢ (𝐹(⇝_𝑢‘𝑆)𝐺 → ∃𝑛 ∈ ℤ 𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆))
28	27	ad2antrr 724	. . 3 ⊢ (((𝐹(⇝_𝑢‘𝑆)𝐺 ∧ 𝐹(⇝_𝑢‘𝑆)𝐻) ∧ 𝑥 ∈ 𝑆) → ∃𝑛 ∈ ℤ 𝐹:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑆))
29	26, 28	r19.29a 3289	. 2 ⊢ (((𝐹(⇝_𝑢‘𝑆)𝐺 ∧ 𝐹(⇝_𝑢‘𝑆)𝐻) ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑥) = (𝐻‘𝑥))
30	3, 6, 29	eqfnfvd 6805	1 ⊢ ((𝐹(⇝_𝑢‘𝑆)𝐺 ∧ 𝐹(⇝_𝑢‘𝑆)𝐻) → 𝐺 = 𝐻)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 Vcvv 3494 class class class wbr 5066 ↦ cmpt 5146 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ↑_m cmap 8406 ℂcc 10535 ℤcz 11982 ℤ_≥cuz 12244 ⇝ cli 14841 ⇝_𝑢culm 24964

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-ulm 24965

This theorem is referenced by: ulmdm 24981

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