cncfmpt2ss - Metamath Proof Explorer

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Theorem cncfmpt2ss 23137

Description: Composition of continuous functions in a subset. (Contributed by Mario Carneiro, 17-May-2016.)

**Hypotheses**
Ref	Expression
cncfmpt2ss.1	⊢ 𝐽 = (TopOpen‘ℂ_fld)
cncfmpt2ss.2	⊢ 𝐹 ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
cncfmpt2ss.3	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→𝑆))
cncfmpt2ss.4	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→𝑆))
cncfmpt2ss.5	⊢ 𝑆 ⊆ ℂ
cncfmpt2ss.6	⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)

**Assertion**
Ref	Expression
cncfmpt2ss	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→𝑆))

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

**Proof of Theorem cncfmpt2ss**
Step	Hyp	Ref	Expression
1		cncfmpt2ss.3	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→𝑆))
2		cncff 23115	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→𝑆) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑆)
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑆)
4	3	fvmptelrn 6649	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑆)
5		cncfmpt2ss.4	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→𝑆))
6		cncff 23115	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→𝑆) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑆)
7	5, 6	syl 17	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑆)
8	7	fvmptelrn 6649	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)
9		cncfmpt2ss.6	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)
10	4, 8, 9	syl2anc 579	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴𝐹𝐵) ∈ 𝑆)
11	10	fmpttd 6651	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)):𝑋⟶𝑆)
12		cncfmpt2ss.5	. . 3 ⊢ 𝑆 ⊆ ℂ
13		cncfmpt2ss.1	. . . 4 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
14		cncfmpt2ss.2	. . . . 5 ⊢ 𝐹 ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
15	14	a1i 11	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
16		ssid 3842	. . . . . 6 ⊢ ℂ ⊆ ℂ
17		cncfss 23121	. . . . . 6 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑋–cn→𝑆) ⊆ (𝑋–cn→ℂ))
18	12, 16, 17	mp2an 682	. . . . 5 ⊢ (𝑋–cn→𝑆) ⊆ (𝑋–cn→ℂ)
19	18, 1	sseldi 3819	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ))
20	18, 5	sseldi 3819	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ))
21	13, 15, 19, 20	cncfmpt2f 23136	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→ℂ))
22		cncffvrn 23120	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→ℂ)) → ((𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→𝑆) ↔ (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)):𝑋⟶𝑆))
23	12, 21, 22	sylancr 581	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→𝑆) ↔ (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)):𝑋⟶𝑆))
24	11, 23	mpbird 249	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ⊆ wss 3792 ↦ cmpt 4967 ⟶wf 6133 ‘cfv 6137 (class class class)co 6924 ℂcc 10272 TopOpenctopn 16479 ℂ_fldccnfld 20153 Cn ccn 21447 ×_t ctx 21783 –cn→ccncf 23098

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fi 8607 df-sup 8638 df-inf 8639 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-z 11734 df-dec 11851 df-uz 11998 df-q 12101 df-rp 12143 df-xneg 12262 df-xadd 12263 df-xmul 12264 df-fz 12649 df-seq 13125 df-exp 13184 df-cj 14252 df-re 14253 df-im 14254 df-sqrt 14388 df-abs 14389 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-plusg 16362 df-mulr 16363 df-starv 16364 df-tset 16368 df-ple 16369 df-ds 16371 df-unif 16372 df-rest 16480 df-topn 16481 df-topgen 16501 df-psmet 20145 df-xmet 20146 df-met 20147 df-bl 20148 df-mopn 20149 df-cnfld 20154 df-top 21117 df-topon 21134 df-topsp 21156 df-bases 21169 df-cn 21450 df-cnp 21451 df-tx 21785 df-xms 22544 df-ms 22545 df-cncf 23100

This theorem is referenced by: cmvth 24202 dvle 24218 dvfsumle 24232 dvfsumge 24233 dvfsumlem2 24238

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