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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 23510 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 2798 | . . . . . . 7 ⊢ (𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) = (𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) | |
3 | 2 | mulc1cncf 23510 | . . . . . 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 23498 | . . . . 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 23498 | . . . . 5 ⊢ (𝐹 ∈ (𝐴–cn→ℂ) → 𝐹:𝐴⟶ℂ) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
10 | fcompt 6872 | . . . 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 587 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) ∘ 𝐹) = (𝑡 ∈ 𝐴 ↦ ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥))‘(𝐹‘𝑡)))) |
12 | 9 | ffvelrnda 6828 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → (𝐹‘𝑡) ∈ ℂ) |
13 | 1 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → 𝐵 ∈ ℂ) |
14 | 13, 12 | mulcld 10650 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → (𝐵 · (𝐹‘𝑡)) ∈ ℂ) |
15 | mulc1cncfg.1 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐹 | |
16 | nfcv 2955 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑡 | |
17 | 15, 16 | nffv 6655 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑡) |
18 | nfcv 2955 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐵 | |
19 | nfcv 2955 | . . . . . . . 8 ⊢ Ⅎ𝑥 · | |
20 | 18, 19, 17 | nfov 7165 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐵 · (𝐹‘𝑡)) |
21 | oveq2 7143 | . . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑡) → (𝐵 · 𝑥) = (𝐵 · (𝐹‘𝑡))) | |
22 | 17, 20, 21, 2 | fvmptf 6766 | . . . . . 6 ⊢ (((𝐹‘𝑡) ∈ ℂ ∧ (𝐵 · (𝐹‘𝑡)) ∈ ℂ) → ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥))‘(𝐹‘𝑡)) = (𝐵 · (𝐹‘𝑡))) |
23 | 12, 14, 22 | syl2anc 587 | . . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥))‘(𝐹‘𝑡)) = (𝐵 · (𝐹‘𝑡))) |
24 | 23 | mpteq2dva 5125 | . . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝐴 ↦ ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥))‘(𝐹‘𝑡))) = (𝑡 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑡)))) |
25 | nfcv 2955 | . . . . . 6 ⊢ Ⅎ𝑡𝐵 | |
26 | nfcv 2955 | . . . . . 6 ⊢ Ⅎ𝑡 · | |
27 | nfcv 2955 | . . . . . 6 ⊢ Ⅎ𝑡(𝐹‘𝑥) | |
28 | 25, 26, 27 | nfov 7165 | . . . . 5 ⊢ Ⅎ𝑡(𝐵 · (𝐹‘𝑥)) |
29 | fveq2 6645 | . . . . . 6 ⊢ (𝑡 = 𝑥 → (𝐹‘𝑡) = (𝐹‘𝑥)) | |
30 | 29 | oveq2d 7151 | . . . . 5 ⊢ (𝑡 = 𝑥 → (𝐵 · (𝐹‘𝑡)) = (𝐵 · (𝐹‘𝑥))) |
31 | 20, 28, 30 | cbvmpt 5131 | . . . 4 ⊢ (𝑡 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑡))) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥))) |
32 | 24, 31 | eqtrdi 2849 | . . 3 ⊢ (𝜑 → (𝑡 ∈ 𝐴 ↦ ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥))‘(𝐹‘𝑡))) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥)))) |
33 | 11, 32 | eqtrd 2833 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥)))) |
34 | 7, 4 | cncfco 23512 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) ∘ 𝐹) ∈ (𝐴–cn→ℂ)) |
35 | 33, 34 | eqeltrrd 2891 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥))) ∈ (𝐴–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 Ⅎwnfc 2936 ↦ cmpt 5110 ∘ ccom 5523 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 · cmul 10531 –cn→ccncf 23481 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-cncf 23483 |
This theorem is referenced by: (None) |
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