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Theorem limcvallem 25829

Description: Lemma for ellimc 25831. (Contributed by Mario Carneiro, 25-Dec-2016.)

**Hypotheses**
Ref	Expression
limcval.j	⊢ 𝐽 = (𝐾 ↾_t (𝐴 ∪ {𝐵}))
limcval.k	⊢ 𝐾 = (TopOpen‘ℂ_fld)
limcvallem.g	⊢ 𝐺 = (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))

**Assertion**
Ref	Expression
limcvallem	⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → (𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵) → 𝐶 ∈ ℂ))

Distinct variable groups: 𝑧,𝐴 𝑧,𝐵 𝑧,𝐹 𝑧,𝐾 𝑧,𝐶 Allowed substitution hints: 𝐺(𝑧) 𝐽(𝑧)

**Proof of Theorem limcvallem**
Step	Hyp	Ref	Expression
1		iftrue 4511	. . . 4 ⊢ (𝑧 = 𝐵 → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) = 𝐶)
2	1	eleq1d 2820	. . 3 ⊢ (𝑧 = 𝐵 → (if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ ℂ ↔ 𝐶 ∈ ℂ))
3		limcval.j	. . . . . 6 ⊢ 𝐽 = (𝐾 ↾_t (𝐴 ∪ {𝐵}))
4		limcval.k	. . . . . . . 8 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
5	4	cnfldtopon 24726	. . . . . . 7 ⊢ 𝐾 ∈ (TopOn‘ℂ)
6		simpl2 1193	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵)) → 𝐴 ⊆ ℂ)
7		simpl3 1194	. . . . . . . . 9 ⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵)) → 𝐵 ∈ ℂ)
8	7	snssd 4790	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵)) → {𝐵} ⊆ ℂ)
9	6, 8	unssd 4172	. . . . . . 7 ⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵)) → (𝐴 ∪ {𝐵}) ⊆ ℂ)
10		resttopon 23104	. . . . . . 7 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ (𝐴 ∪ {𝐵}) ⊆ ℂ) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
11	5, 9, 10	sylancr 587	. . . . . 6 ⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵)) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
12	3, 11	eqeltrid 2839	. . . . 5 ⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵)) → 𝐽 ∈ (TopOn‘(𝐴 ∪ {𝐵})))
13	5	a1i 11	. . . . 5 ⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵)) → 𝐾 ∈ (TopOn‘ℂ))
14		simpr 484	. . . . 5 ⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵)) → 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵))
15		cnpf2 23193	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘(𝐴 ∪ {𝐵})) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵)) → 𝐺:(𝐴 ∪ {𝐵})⟶ℂ)
16	12, 13, 14, 15	syl3anc 1373	. . . 4 ⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵)) → 𝐺:(𝐴 ∪ {𝐵})⟶ℂ)
17		limcvallem.g	. . . . 5 ⊢ 𝐺 = (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))
18	17	fmpt 7105	. . . 4 ⊢ (∀𝑧 ∈ (𝐴 ∪ {𝐵})if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ ℂ ↔ 𝐺:(𝐴 ∪ {𝐵})⟶ℂ)
19	16, 18	sylibr 234	. . 3 ⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵)) → ∀𝑧 ∈ (𝐴 ∪ {𝐵})if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ ℂ)
20		ssun2 4159	. . . 4 ⊢ {𝐵} ⊆ (𝐴 ∪ {𝐵})
21		snssg 4764	. . . . 5 ⊢ (𝐵 ∈ ℂ → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
22	7, 21	syl 17	. . . 4 ⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵)) → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
23	20, 22	mpbiri 258	. . 3 ⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵)) → 𝐵 ∈ (𝐴 ∪ {𝐵}))
24	2, 19, 23	rspcdva 3607	. 2 ⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵)) → 𝐶 ∈ ℂ)
25	24	ex 412	1 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → (𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵) → 𝐶 ∈ ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3052 ∪ cun 3929 ⊆ wss 3931 ifcif 4505 {csn 4606 ↦ cmpt 5206 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ℂcc 11132 ↾_t crest 17439 TopOpenctopn 17440 ℂ_fldccnfld 21320 TopOnctopon 22853 CnP ccnp 23168

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 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212

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 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fi 9428 df-sup 9459 df-inf 9460 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-fz 13530 df-seq 14025 df-exp 14085 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-struct 17171 df-slot 17206 df-ndx 17218 df-base 17234 df-plusg 17289 df-mulr 17290 df-starv 17291 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-rest 17441 df-topn 17442 df-topgen 17462 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-cnfld 21321 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cnp 23171 df-xms 24264 df-ms 24265

This theorem is referenced by: limcfval 25830 ellimc 25831

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