cncfmpt2ss - Metamath Proof Explorer

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Theorem cncfmpt2ss 24079

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 24056	. . . . . 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 6987	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑆)
5		cncfmpt2ss.4	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→𝑆))
6		cncff 24056	. . . . . 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 6987	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)
9		cncfmpt2ss.6	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)
10	4, 8, 9	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴𝐹𝐵) ∈ 𝑆)
11	10	fmpttd 6989	. 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 3943	. . . . . 6 ⊢ ℂ ⊆ ℂ
17		cncfss 24062	. . . . . 6 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑋–cn→𝑆) ⊆ (𝑋–cn→ℂ))
18	12, 16, 17	mp2an 689	. . . . 5 ⊢ (𝑋–cn→𝑆) ⊆ (𝑋–cn→ℂ)
19	18, 1	sselid 3919	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ))
20	18, 5	sselid 3919	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ))
21	13, 15, 19, 20	cncfmpt2f 24078	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→ℂ))
22		cncffvrn 24061	. . 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 587	. 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 396 = wceq 1539 ∈ wcel 2106 ⊆ wss 3887 ↦ cmpt 5157 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 TopOpenctopn 17132 ℂ_fldccnfld 20597 Cn ccn 22375 ×_t ctx 22711 –cn→ccncf 24039

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fi 9170 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-fz 13240 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-struct 16848 df-slot 16883 df-ndx 16895 df-base 16913 df-plusg 16975 df-mulr 16976 df-starv 16977 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-rest 17133 df-topn 17134 df-topgen 17154 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cn 22378 df-cnp 22379 df-tx 22713 df-xms 23473 df-ms 23474 df-cncf 24041

This theorem is referenced by: cmvth 25155 dvle 25171 dvfsumle 25185 dvfsumge 25186 dvfsumlem2 25191

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