Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > mulc1cncfg | Structured version Visualization version GIF version |
Description: A version of mulc1cncf 23440 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 2818 | . . . . . . 7 ⊢ (𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) = (𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) | |
3 | 2 | mulc1cncf 23440 | . . . . . 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 23428 | . . . . 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 23428 | . . . . 5 ⊢ (𝐹 ∈ (𝐴–cn→ℂ) → 𝐹:𝐴⟶ℂ) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
10 | fcompt 6887 | . . . 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 584 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) ∘ 𝐹) = (𝑡 ∈ 𝐴 ↦ ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥))‘(𝐹‘𝑡)))) |
12 | 9 | ffvelrnda 6843 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → (𝐹‘𝑡) ∈ ℂ) |
13 | 1 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → 𝐵 ∈ ℂ) |
14 | 13, 12 | mulcld 10649 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → (𝐵 · (𝐹‘𝑡)) ∈ ℂ) |
15 | mulc1cncfg.1 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐹 | |
16 | nfcv 2974 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑡 | |
17 | 15, 16 | nffv 6673 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑡) |
18 | nfcv 2974 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐵 | |
19 | nfcv 2974 | . . . . . . . 8 ⊢ Ⅎ𝑥 · | |
20 | 18, 19, 17 | nfov 7175 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐵 · (𝐹‘𝑡)) |
21 | oveq2 7153 | . . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑡) → (𝐵 · 𝑥) = (𝐵 · (𝐹‘𝑡))) | |
22 | 17, 20, 21, 2 | fvmptf 6781 | . . . . . 6 ⊢ (((𝐹‘𝑡) ∈ ℂ ∧ (𝐵 · (𝐹‘𝑡)) ∈ ℂ) → ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥))‘(𝐹‘𝑡)) = (𝐵 · (𝐹‘𝑡))) |
23 | 12, 14, 22 | syl2anc 584 | . . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥))‘(𝐹‘𝑡)) = (𝐵 · (𝐹‘𝑡))) |
24 | 23 | mpteq2dva 5152 | . . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝐴 ↦ ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥))‘(𝐹‘𝑡))) = (𝑡 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑡)))) |
25 | nfcv 2974 | . . . . . 6 ⊢ Ⅎ𝑡𝐵 | |
26 | nfcv 2974 | . . . . . 6 ⊢ Ⅎ𝑡 · | |
27 | nfcv 2974 | . . . . . 6 ⊢ Ⅎ𝑡(𝐹‘𝑥) | |
28 | 25, 26, 27 | nfov 7175 | . . . . 5 ⊢ Ⅎ𝑡(𝐵 · (𝐹‘𝑥)) |
29 | fveq2 6663 | . . . . . 6 ⊢ (𝑡 = 𝑥 → (𝐹‘𝑡) = (𝐹‘𝑥)) | |
30 | 29 | oveq2d 7161 | . . . . 5 ⊢ (𝑡 = 𝑥 → (𝐵 · (𝐹‘𝑡)) = (𝐵 · (𝐹‘𝑥))) |
31 | 20, 28, 30 | cbvmpt 5158 | . . . 4 ⊢ (𝑡 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑡))) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥))) |
32 | 24, 31 | syl6eq 2869 | . . 3 ⊢ (𝜑 → (𝑡 ∈ 𝐴 ↦ ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥))‘(𝐹‘𝑡))) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥)))) |
33 | 11, 32 | eqtrd 2853 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥)))) |
34 | 7, 4 | cncfco 23442 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) ∘ 𝐹) ∈ (𝐴–cn→ℂ)) |
35 | 33, 34 | eqeltrrd 2911 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥))) ∈ (𝐴–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 Ⅎwnfc 2958 ↦ cmpt 5137 ∘ ccom 5552 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 · cmul 10530 –cn→ccncf 23411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-cncf 23413 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |