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Theorem cnmptre 24854

Description: Lemma for iirevcn 24857 and related functions. (Contributed by Mario Carneiro, 6-Jun-2014.)

**Hypotheses**
Ref	Expression
cnmptre.1	⊢ 𝑅 = (TopOpen‘ℂ_fld)
cnmptre.2	⊢ 𝐽 = ((topGen‘ran (,)) ↾_t 𝐴)
cnmptre.3	⊢ 𝐾 = ((topGen‘ran (,)) ↾_t 𝐵)
cnmptre.4	⊢ (𝜑 → 𝐴 ⊆ ℝ)
cnmptre.5	⊢ (𝜑 → 𝐵 ⊆ ℝ)
cnmptre.6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ 𝐵)
cnmptre.7	⊢ (𝜑 → (𝑥 ∈ ℂ ↦ 𝐹) ∈ (𝑅 Cn 𝑅))

**Assertion**
Ref	Expression
cnmptre	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn 𝐾))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝜑,𝑥 Allowed substitution hints: 𝑅(𝑥) 𝐹(𝑥) 𝐽(𝑥) 𝐾(𝑥)

**Proof of Theorem cnmptre**
Step	Hyp	Ref	Expression
1		eqid 2729	. . . . 5 ⊢ (𝑅 ↾_t 𝐴) = (𝑅 ↾_t 𝐴)
2		cnmptre.1	. . . . . . 7 ⊢ 𝑅 = (TopOpen‘ℂ_fld)
3	2	cnfldtopon 24703	. . . . . 6 ⊢ 𝑅 ∈ (TopOn‘ℂ)
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (TopOn‘ℂ))
5		cnmptre.4	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
6		ax-resscn 11101	. . . . . 6 ⊢ ℝ ⊆ ℂ
7	5, 6	sstrdi 3956	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
8		cnmptre.7	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ 𝐹) ∈ (𝑅 Cn 𝑅))
9	1, 4, 7, 8	cnmpt1res 23596	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ ((𝑅 ↾_t 𝐴) Cn 𝑅))
10		eqid 2729	. . . . . . . 8 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
11	2, 10	rerest 24725	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ → (𝑅 ↾_t 𝐴) = ((topGen‘ran (,)) ↾_t 𝐴))
12	5, 11	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑅 ↾_t 𝐴) = ((topGen‘ran (,)) ↾_t 𝐴))
13		cnmptre.2	. . . . . 6 ⊢ 𝐽 = ((topGen‘ran (,)) ↾_t 𝐴)
14	12, 13	eqtr4di 2782	. . . . 5 ⊢ (𝜑 → (𝑅 ↾_t 𝐴) = 𝐽)
15	14	oveq1d 7384	. . . 4 ⊢ (𝜑 → ((𝑅 ↾_t 𝐴) Cn 𝑅) = (𝐽 Cn 𝑅))
16	9, 15	eleqtrd 2830	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn 𝑅))
17		cnmptre.6	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ 𝐵)
18	17	fmpttd 7069	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹):𝐴⟶𝐵)
19	18	frnd 6678	. . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐹) ⊆ 𝐵)
20		cnmptre.5	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
21	20, 6	sstrdi 3956	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℂ)
22		cnrest2 23206	. . . 4 ⊢ ((𝑅 ∈ (TopOn‘ℂ) ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐹) ⊆ 𝐵 ∧ 𝐵 ⊆ ℂ) → ((𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn 𝑅) ↔ (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn (𝑅 ↾_t 𝐵))))
23	3, 19, 21, 22	mp3an2i 1468	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn 𝑅) ↔ (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn (𝑅 ↾_t 𝐵))))
24	16, 23	mpbid 232	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn (𝑅 ↾_t 𝐵)))
25	2, 10	rerest 24725	. . . . 5 ⊢ (𝐵 ⊆ ℝ → (𝑅 ↾_t 𝐵) = ((topGen‘ran (,)) ↾_t 𝐵))
26	20, 25	syl 17	. . . 4 ⊢ (𝜑 → (𝑅 ↾_t 𝐵) = ((topGen‘ran (,)) ↾_t 𝐵))
27		cnmptre.3	. . . 4 ⊢ 𝐾 = ((topGen‘ran (,)) ↾_t 𝐵)
28	26, 27	eqtr4di 2782	. . 3 ⊢ (𝜑 → (𝑅 ↾_t 𝐵) = 𝐾)
29	28	oveq2d 7385	. 2 ⊢ (𝜑 → (𝐽 Cn (𝑅 ↾_t 𝐵)) = (𝐽 Cn 𝐾))
30	24, 29	eleqtrd 2830	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ⊆ wss 3911 ↦ cmpt 5183 ran crn 5632 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 ℝcr 11043 (,)cioo 13282 ↾_t crest 17359 TopOpenctopn 17360 topGenctg 17376 ℂ_fldccnfld 21296 TopOnctopon 22830 Cn ccn 23144

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 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122

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 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-er 8648 df-map 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fi 9338 df-sup 9369 df-inf 9370 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-fz 13445 df-seq 13943 df-exp 14003 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-struct 17093 df-slot 17128 df-ndx 17140 df-base 17156 df-plusg 17209 df-mulr 17210 df-starv 17211 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-rest 17361 df-topn 17362 df-topgen 17382 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-cnfld 21297 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22866 df-cn 23147 df-xms 24241 df-ms 24242

This theorem is referenced by: iirevcn 24857 iihalf1cn 24859 iihalf1cnOLD 24860 iihalf2cn 24862 iihalf2cnOLD 24863 pcoass 24957

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