Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mulc1cncfg | Structured version Visualization version GIF version |
Description: A version of mulc1cncf 22708 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 30-Jun-2017.) |
Ref | Expression |
---|---|
mulc1cncfg.1 | ⊢ Ⅎ𝑥𝐹 |
mulc1cncfg.2 | ⊢ Ⅎ𝑥𝜑 |
mulc1cncfg.3 | ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ)) |
mulc1cncfg.4 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
mulc1cncfg | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥))) ∈ (𝐴–cn→ℂ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulc1cncfg.4 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
2 | eqid 2622 | . . . . . . 7 ⊢ (𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) = (𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) | |
3 | 2 | mulc1cncf 22708 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → (𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) ∈ (ℂ–cn→ℂ)) |
4 | 1, 3 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) ∈ (ℂ–cn→ℂ)) |
5 | cncff 22696 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) ∈ (ℂ–cn→ℂ) → (𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)):ℂ⟶ℂ) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)):ℂ⟶ℂ) |
7 | mulc1cncfg.3 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ)) | |
8 | cncff 22696 | . . . . 5 ⊢ (𝐹 ∈ (𝐴–cn→ℂ) → 𝐹:𝐴⟶ℂ) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
10 | fcompt 6400 | . . . 4 ⊢ (((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)):ℂ⟶ℂ ∧ 𝐹:𝐴⟶ℂ) → ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) ∘ 𝐹) = (𝑡 ∈ 𝐴 ↦ ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥))‘(𝐹‘𝑡)))) | |
11 | 6, 9, 10 | syl2anc 693 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) ∘ 𝐹) = (𝑡 ∈ 𝐴 ↦ ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥))‘(𝐹‘𝑡)))) |
12 | 9 | ffvelrnda 6359 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → (𝐹‘𝑡) ∈ ℂ) |
13 | 1 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → 𝐵 ∈ ℂ) |
14 | 13, 12 | mulcld 10060 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → (𝐵 · (𝐹‘𝑡)) ∈ ℂ) |
15 | mulc1cncfg.1 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐹 | |
16 | nfcv 2764 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑡 | |
17 | 15, 16 | nffv 6198 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑡) |
18 | nfcv 2764 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐵 | |
19 | nfcv 2764 | . . . . . . . 8 ⊢ Ⅎ𝑥 · | |
20 | 18, 19, 17 | nfov 6676 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐵 · (𝐹‘𝑡)) |
21 | oveq2 6658 | . . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑡) → (𝐵 · 𝑥) = (𝐵 · (𝐹‘𝑡))) | |
22 | 17, 20, 21, 2 | fvmptf 6301 | . . . . . 6 ⊢ (((𝐹‘𝑡) ∈ ℂ ∧ (𝐵 · (𝐹‘𝑡)) ∈ ℂ) → ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥))‘(𝐹‘𝑡)) = (𝐵 · (𝐹‘𝑡))) |
23 | 12, 14, 22 | syl2anc 693 | . . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥))‘(𝐹‘𝑡)) = (𝐵 · (𝐹‘𝑡))) |
24 | 23 | mpteq2dva 4744 | . . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝐴 ↦ ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥))‘(𝐹‘𝑡))) = (𝑡 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑡)))) |
25 | nfcv 2764 | . . . . . 6 ⊢ Ⅎ𝑡𝐵 | |
26 | nfcv 2764 | . . . . . 6 ⊢ Ⅎ𝑡 · | |
27 | nfcv 2764 | . . . . . 6 ⊢ Ⅎ𝑡(𝐹‘𝑥) | |
28 | 25, 26, 27 | nfov 6676 | . . . . 5 ⊢ Ⅎ𝑡(𝐵 · (𝐹‘𝑥)) |
29 | fveq2 6191 | . . . . . 6 ⊢ (𝑡 = 𝑥 → (𝐹‘𝑡) = (𝐹‘𝑥)) | |
30 | 29 | oveq2d 6666 | . . . . 5 ⊢ (𝑡 = 𝑥 → (𝐵 · (𝐹‘𝑡)) = (𝐵 · (𝐹‘𝑥))) |
31 | 20, 28, 30 | cbvmpt 4749 | . . . 4 ⊢ (𝑡 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑡))) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥))) |
32 | 24, 31 | syl6eq 2672 | . . 3 ⊢ (𝜑 → (𝑡 ∈ 𝐴 ↦ ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥))‘(𝐹‘𝑡))) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥)))) |
33 | 11, 32 | eqtrd 2656 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥)))) |
34 | 7, 4 | cncfco 22710 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) ∘ 𝐹) ∈ (𝐴–cn→ℂ)) |
35 | 33, 34 | eqeltrrd 2702 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥))) ∈ (𝐴–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 Ⅎwnfc 2751 ↦ cmpt 4729 ∘ ccom 5118 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 · cmul 9941 –cn→ccncf 22679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-cncf 22681 |
This theorem is referenced by: (None) |
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