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Theorem cnmptre 24973

Description: Lemma for iirevcn 24976 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 2740	. . . . 5 ⊢ (𝑅 ↾_t 𝐴) = (𝑅 ↾_t 𝐴)
2		cnmptre.1	. . . . . . 7 ⊢ 𝑅 = (TopOpen‘ℂ_fld)
3	2	cnfldtopon 24824	. . . . . 6 ⊢ 𝑅 ∈ (TopOn‘ℂ)
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (TopOn‘ℂ))
5		cnmptre.4	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
6		ax-resscn 11241	. . . . . 6 ⊢ ℝ ⊆ ℂ
7	5, 6	sstrdi 4021	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
8		cnmptre.7	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ 𝐹) ∈ (𝑅 Cn 𝑅))
9	1, 4, 7, 8	cnmpt1res 23705	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ ((𝑅 ↾_t 𝐴) Cn 𝑅))
10		eqid 2740	. . . . . . . 8 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
11	2, 10	rerest 24845	. . . . . . 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 2798	. . . . 5 ⊢ (𝜑 → (𝑅 ↾_t 𝐴) = 𝐽)
15	14	oveq1d 7463	. . . 4 ⊢ (𝜑 → ((𝑅 ↾_t 𝐴) Cn 𝑅) = (𝐽 Cn 𝑅))
16	9, 15	eleqtrd 2846	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn 𝑅))
17		cnmptre.6	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ 𝐵)
18	17	fmpttd 7149	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹):𝐴⟶𝐵)
19	18	frnd 6755	. . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐹) ⊆ 𝐵)
20		cnmptre.5	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
21	20, 6	sstrdi 4021	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℂ)
22		cnrest2 23315	. . . 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 1466	. . 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 24845	. . . . 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 2798	. . 3 ⊢ (𝜑 → (𝑅 ↾_t 𝐵) = 𝐾)
29	28	oveq2d 7464	. 2 ⊢ (𝜑 → (𝐽 Cn (𝑅 ↾_t 𝐵)) = (𝐽 Cn 𝐾))
30	24, 29	eleqtrd 2846	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ⊆ wss 3976 ↦ cmpt 5249 ran crn 5701 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 (,)cioo 13407 ↾_t crest 17480 TopOpenctopn 17481 topGenctg 17497 ℂ_fldccnfld 21387 TopOnctopon 22937 Cn ccn 23253

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fi 9480 df-sup 9511 df-inf 9512 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-fz 13568 df-seq 14053 df-exp 14113 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-struct 17194 df-slot 17229 df-ndx 17241 df-base 17259 df-plusg 17324 df-mulr 17325 df-starv 17326 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-rest 17482 df-topn 17483 df-topgen 17503 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cn 23256 df-xms 24351 df-ms 24352

This theorem is referenced by: iirevcn 24976 iihalf1cn 24978 iihalf1cnOLD 24979 iihalf2cn 24981 iihalf2cnOLD 24982 pcoass 25076

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