cncfmpt2ss - Metamath Proof Explorer

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Theorem cncfmpt2ss 23985

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 23962	. . . . . 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 6969	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑆)
5		cncfmpt2ss.4	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→𝑆))
6		cncff 23962	. . . . . 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 6969	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)
9		cncfmpt2ss.6	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)
10	4, 8, 9	syl2anc 583	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴𝐹𝐵) ∈ 𝑆)
11	10	fmpttd 6971	. 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 3939	. . . . . 6 ⊢ ℂ ⊆ ℂ
17		cncfss 23968	. . . . . 6 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑋–cn→𝑆) ⊆ (𝑋–cn→ℂ))
18	12, 16, 17	mp2an 688	. . . . 5 ⊢ (𝑋–cn→𝑆) ⊆ (𝑋–cn→ℂ)
19	18, 1	sselid 3915	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ))
20	18, 5	sselid 3915	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ))
21	13, 15, 19, 20	cncfmpt2f 23984	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→ℂ))
22		cncffvrn 23967	. . 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 586	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→𝑆) ↔ (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)):𝑋⟶𝑆))
24	11, 23	mpbird 256	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ⊆ wss 3883 ↦ cmpt 5153 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 TopOpenctopn 17049 ℂ_fldccnfld 20510 Cn ccn 22283 ×_t ctx 22619 –cn→ccncf 23945

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fi 9100 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-fz 13169 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-struct 16776 df-slot 16811 df-ndx 16823 df-base 16841 df-plusg 16901 df-mulr 16902 df-starv 16903 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-rest 17050 df-topn 17051 df-topgen 17071 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cn 22286 df-cnp 22287 df-tx 22621 df-xms 23381 df-ms 23382 df-cncf 23947

This theorem is referenced by: cmvth 25060 dvle 25076 dvfsumle 25090 dvfsumge 25091 dvfsumlem2 25096

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