cncfmpt2ss - Metamath Proof Explorer

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Theorem cncfmpt2ss 24883

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 24860	. . . . . 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	fvmptelcdm 7065	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑆)
5		cncfmpt2ss.4	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→𝑆))
6		cncff 24860	. . . . . 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	fvmptelcdm 7065	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)
9		cncfmpt2ss.6	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)
10	4, 8, 9	syl2anc 585	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴𝐹𝐵) ∈ 𝑆)
11	10	fmpttd 7067	. 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 3944	. . . . . 6 ⊢ ℂ ⊆ ℂ
17		cncfss 24866	. . . . . 6 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑋–cn→𝑆) ⊆ (𝑋–cn→ℂ))
18	12, 16, 17	mp2an 693	. . . . 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 24882	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→ℂ))
22		cncfcdm 24865	. . 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 588	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→𝑆) ↔ (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)):𝑋⟶𝑆))
24	11, 23	mpbird 257	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ⊆ wss 3889 ↦ cmpt 5166 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 TopOpenctopn 17384 ℂ_fldccnfld 21352 Cn ccn 23189 ×_t ctx 23525 –cn→ccncf 24843

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fi 9324 df-sup 9355 df-inf 9356 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-fz 13462 df-seq 13964 df-exp 14024 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-struct 17117 df-slot 17152 df-ndx 17164 df-base 17180 df-plusg 17233 df-mulr 17234 df-starv 17235 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-rest 17385 df-topn 17386 df-topgen 17406 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cn 23192 df-cnp 23193 df-tx 23527 df-xms 24285 df-ms 24286 df-cncf 24845

This theorem is referenced by: cmvth 25958 dvle 25974 dvfsumle 25988 dvfsumge 25989 dvfsumlem2 25994

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