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Mirrors > Home > MPE Home > Th. List > cncfmpt2ss | Structured version Visualization version GIF version |
Description: Composition of continuous functions in a subset. (Contributed by Mario Carneiro, 17-May-2016.) |
Ref | Expression |
---|---|
cncfmpt2ss.1 | ⊢ 𝐽 = (TopOpen‘ℂfld) |
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) |
Ref | Expression |
---|---|
cncfmpt2ss | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncfmpt2ss.3 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→𝑆)) | |
2 | cncff 22504 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→𝑆) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑆) | |
3 | 1, 2 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑆) |
4 | eqid 2610 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴) | |
5 | 4 | fmpt 6289 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑆 ↔ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑆) |
6 | 3, 5 | sylibr 223 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑆) |
7 | 6 | r19.21bi 2916 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑆) |
8 | cncfmpt2ss.4 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→𝑆)) | |
9 | cncff 22504 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→𝑆) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑆) | |
10 | 8, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑆) |
11 | eqid 2610 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵) | |
12 | 11 | fmpt 6289 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆 ↔ (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑆) |
13 | 10, 12 | sylibr 223 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆) |
14 | 13 | r19.21bi 2916 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆) |
15 | cncfmpt2ss.6 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆) | |
16 | 7, 14, 15 | syl2anc 691 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴𝐹𝐵) ∈ 𝑆) |
17 | eqid 2610 | . . 3 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) | |
18 | 16, 17 | fmptd 6292 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)):𝑋⟶𝑆) |
19 | cncfmpt2ss.5 | . . 3 ⊢ 𝑆 ⊆ ℂ | |
20 | cncfmpt2ss.1 | . . . 4 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
21 | cncfmpt2ss.2 | . . . . 5 ⊢ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽) | |
22 | 21 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
23 | ssid 3587 | . . . . . 6 ⊢ ℂ ⊆ ℂ | |
24 | cncfss 22510 | . . . . . 6 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑋–cn→𝑆) ⊆ (𝑋–cn→ℂ)) | |
25 | 19, 23, 24 | mp2an 704 | . . . . 5 ⊢ (𝑋–cn→𝑆) ⊆ (𝑋–cn→ℂ) |
26 | 25, 1 | sseldi 3566 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ)) |
27 | 25, 8 | sseldi 3566 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ)) |
28 | 20, 22, 26, 27 | cncfmpt2f 22525 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→ℂ)) |
29 | cncffvrn 22509 | . . 3 ⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→ℂ)) → ((𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→𝑆) ↔ (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)):𝑋⟶𝑆)) | |
30 | 19, 28, 29 | sylancr 694 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→𝑆) ↔ (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)):𝑋⟶𝑆)) |
31 | 18, 30 | mpbird 246 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 ↦ cmpt 4643 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 TopOpenctopn 15905 ℂfldccnfld 19567 Cn ccn 20838 ×t ctx 21173 –cn→ccncf 22487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-rest 15906 df-topn 15907 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cn 20841 df-cnp 20842 df-tx 21175 df-xms 21935 df-ms 21936 df-cncf 22489 |
This theorem is referenced by: cmvth 23558 dvle 23574 dvfsumle 23588 dvfsumge 23589 dvfsumlem2 23594 |
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