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Theorem cnmptre 24968

Description: Lemma for iirevcn 24971 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 2735	. . . . 5 ⊢ (𝑅 ↾_t 𝐴) = (𝑅 ↾_t 𝐴)
2		cnmptre.1	. . . . . . 7 ⊢ 𝑅 = (TopOpen‘ℂ_fld)
3	2	cnfldtopon 24819	. . . . . 6 ⊢ 𝑅 ∈ (TopOn‘ℂ)
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (TopOn‘ℂ))
5		cnmptre.4	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
6		ax-resscn 11210	. . . . . 6 ⊢ ℝ ⊆ ℂ
7	5, 6	sstrdi 4008	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
8		cnmptre.7	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ 𝐹) ∈ (𝑅 Cn 𝑅))
9	1, 4, 7, 8	cnmpt1res 23700	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ ((𝑅 ↾_t 𝐴) Cn 𝑅))
10		eqid 2735	. . . . . . . 8 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
11	2, 10	rerest 24840	. . . . . . 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 2793	. . . . 5 ⊢ (𝜑 → (𝑅 ↾_t 𝐴) = 𝐽)
15	14	oveq1d 7446	. . . 4 ⊢ (𝜑 → ((𝑅 ↾_t 𝐴) Cn 𝑅) = (𝐽 Cn 𝑅))
16	9, 15	eleqtrd 2841	. . 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 6745	. . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐹) ⊆ 𝐵)
20		cnmptre.5	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
21	20, 6	sstrdi 4008	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℂ)
22		cnrest2 23310	. . . 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 1465	. . 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 24840	. . . . 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 2793	. . 3 ⊢ (𝜑 → (𝑅 ↾_t 𝐵) = 𝐾)
29	28	oveq2d 7447	. 2 ⊢ (𝜑 → (𝐽 Cn (𝑅 ↾_t 𝐵)) = (𝐽 Cn 𝐾))
30	24, 29	eleqtrd 2841	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 2106 ⊆ wss 3963 ↦ cmpt 5231 ran crn 5690 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 (,)cioo 13384 ↾_t crest 17467 TopOpenctopn 17468 topGenctg 17484 ℂ_fldccnfld 21382 TopOnctopon 22932 Cn ccn 23248

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fi 9449 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-fz 13545 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-starv 17313 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-rest 17469 df-topn 17470 df-topgen 17490 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cn 23251 df-xms 24346 df-ms 24347

This theorem is referenced by: iirevcn 24971 iihalf1cn 24973 iihalf1cnOLD 24974 iihalf2cn 24976 iihalf2cnOLD 24977 pcoass 25071

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