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Theorem cnmptre 24954

Description: Lemma for iirevcn 24957 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 2737	. . . . 5 ⊢ (𝑅 ↾_t 𝐴) = (𝑅 ↾_t 𝐴)
2		cnmptre.1	. . . . . . 7 ⊢ 𝑅 = (TopOpen‘ℂ_fld)
3	2	cnfldtopon 24803	. . . . . 6 ⊢ 𝑅 ∈ (TopOn‘ℂ)
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (TopOn‘ℂ))
5		cnmptre.4	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
6		ax-resscn 11212	. . . . . 6 ⊢ ℝ ⊆ ℂ
7	5, 6	sstrdi 3996	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
8		cnmptre.7	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ 𝐹) ∈ (𝑅 Cn 𝑅))
9	1, 4, 7, 8	cnmpt1res 23684	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ ((𝑅 ↾_t 𝐴) Cn 𝑅))
10		eqid 2737	. . . . . . . 8 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
11	2, 10	rerest 24825	. . . . . . 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 2795	. . . . 5 ⊢ (𝜑 → (𝑅 ↾_t 𝐴) = 𝐽)
15	14	oveq1d 7446	. . . 4 ⊢ (𝜑 → ((𝑅 ↾_t 𝐴) Cn 𝑅) = (𝐽 Cn 𝑅))
16	9, 15	eleqtrd 2843	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn 𝑅))
17		cnmptre.6	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ 𝐵)
18	17	fmpttd 7135	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹):𝐴⟶𝐵)
19	18	frnd 6744	. . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐹) ⊆ 𝐵)
20		cnmptre.5	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
21	20, 6	sstrdi 3996	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℂ)
22		cnrest2 23294	. . . 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 24825	. . . . 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 2795	. . 3 ⊢ (𝜑 → (𝑅 ↾_t 𝐵) = 𝐾)
29	28	oveq2d 7447	. 2 ⊢ (𝜑 → (𝐽 Cn (𝑅 ↾_t 𝐵)) = (𝐽 Cn 𝐾))
30	24, 29	eleqtrd 2843	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 2108 ⊆ wss 3951 ↦ cmpt 5225 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 (,)cioo 13387 ↾_t crest 17465 TopOpenctopn 17466 topGenctg 17482 ℂ_fldccnfld 21364 TopOnctopon 22916 Cn ccn 23232

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-rest 17467 df-topn 17468 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cn 23235 df-xms 24330 df-ms 24331

This theorem is referenced by: iirevcn 24957 iihalf1cn 24959 iihalf1cnOLD 24960 iihalf2cn 24962 iihalf2cnOLD 24963 pcoass 25057

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