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Theorem cnmptre 24879

Description: Lemma for iirevcn 24882 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 2736	. . . . 5 ⊢ (𝑅 ↾_t 𝐴) = (𝑅 ↾_t 𝐴)
2		cnmptre.1	. . . . . . 7 ⊢ 𝑅 = (TopOpen‘ℂ_fld)
3	2	cnfldtopon 24728	. . . . . 6 ⊢ 𝑅 ∈ (TopOn‘ℂ)
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (TopOn‘ℂ))
5		cnmptre.4	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
6		ax-resscn 11085	. . . . . 6 ⊢ ℝ ⊆ ℂ
7	5, 6	sstrdi 3946	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
8		cnmptre.7	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ 𝐹) ∈ (𝑅 Cn 𝑅))
9	1, 4, 7, 8	cnmpt1res 23622	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ ((𝑅 ↾_t 𝐴) Cn 𝑅))
10		eqid 2736	. . . . . . . 8 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
11	2, 10	rerest 24750	. . . . . . 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 2789	. . . . 5 ⊢ (𝜑 → (𝑅 ↾_t 𝐴) = 𝐽)
15	14	oveq1d 7373	. . . 4 ⊢ (𝜑 → ((𝑅 ↾_t 𝐴) Cn 𝑅) = (𝐽 Cn 𝑅))
16	9, 15	eleqtrd 2838	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn 𝑅))
17		cnmptre.6	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ 𝐵)
18	17	fmpttd 7060	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹):𝐴⟶𝐵)
19	18	frnd 6670	. . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐹) ⊆ 𝐵)
20		cnmptre.5	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
21	20, 6	sstrdi 3946	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℂ)
22		cnrest2 23232	. . . 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 24750	. . . . 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 2789	. . 3 ⊢ (𝜑 → (𝑅 ↾_t 𝐵) = 𝐾)
29	28	oveq2d 7374	. 2 ⊢ (𝜑 → (𝐽 Cn (𝑅 ↾_t 𝐵)) = (𝐽 Cn 𝐾))
30	24, 29	eleqtrd 2838	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ⊆ wss 3901 ↦ cmpt 5179 ran crn 5625 ‘cfv 6492 (class class class)co 7358 ℂcc 11026 ℝcr 11027 (,)cioo 13263 ↾_t crest 17342 TopOpenctopn 17343 topGenctg 17359 ℂ_fldccnfld 21311 TopOnctopon 22856 Cn ccn 23170

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-map 8767 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-fi 9316 df-sup 9347 df-inf 9348 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12610 df-uz 12754 df-q 12864 df-rp 12908 df-xneg 13028 df-xadd 13029 df-xmul 13030 df-ioo 13267 df-fz 13426 df-seq 13927 df-exp 13987 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17139 df-plusg 17192 df-mulr 17193 df-starv 17194 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-rest 17344 df-topn 17345 df-topgen 17365 df-psmet 21303 df-xmet 21304 df-met 21305 df-bl 21306 df-mopn 21307 df-cnfld 21312 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22892 df-cn 23173 df-xms 24266 df-ms 24267

This theorem is referenced by: iirevcn 24882 iihalf1cn 24884 iihalf1cnOLD 24885 iihalf2cn 24887 iihalf2cnOLD 24888 pcoass 24982

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