|   | 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 24932 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 2736 | . . . . . . 7 ⊢ (𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) = (𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) | |
| 3 | 2 | mulc1cncf 24932 | . . . . . 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 24920 | . . . . 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 24920 | . . . . 5 ⊢ (𝐹 ∈ (𝐴–cn→ℂ) → 𝐹:𝐴⟶ℂ) | |
| 9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | 
| 10 | fcompt 7152 | . . . 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 | ffvelcdmda 7103 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → (𝐹‘𝑡) ∈ ℂ) | 
| 13 | 1 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → 𝐵 ∈ ℂ) | 
| 14 | 13, 12 | mulcld 11282 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → (𝐵 · (𝐹‘𝑡)) ∈ ℂ) | 
| 15 | mulc1cncfg.1 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐹 | |
| 16 | nfcv 2904 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑡 | |
| 17 | 15, 16 | nffv 6915 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑡) | 
| 18 | nfcv 2904 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐵 | |
| 19 | nfcv 2904 | . . . . . . . 8 ⊢ Ⅎ𝑥 · | |
| 20 | 18, 19, 17 | nfov 7462 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐵 · (𝐹‘𝑡)) | 
| 21 | oveq2 7440 | . . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑡) → (𝐵 · 𝑥) = (𝐵 · (𝐹‘𝑡))) | |
| 22 | 17, 20, 21, 2 | fvmptf 7036 | . . . . . 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 5241 | . . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝐴 ↦ ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥))‘(𝐹‘𝑡))) = (𝑡 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑡)))) | 
| 25 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑡𝐵 | |
| 26 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑡 · | |
| 27 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑡(𝐹‘𝑥) | |
| 28 | 25, 26, 27 | nfov 7462 | . . . . 5 ⊢ Ⅎ𝑡(𝐵 · (𝐹‘𝑥)) | 
| 29 | fveq2 6905 | . . . . . 6 ⊢ (𝑡 = 𝑥 → (𝐹‘𝑡) = (𝐹‘𝑥)) | |
| 30 | 29 | oveq2d 7448 | . . . . 5 ⊢ (𝑡 = 𝑥 → (𝐵 · (𝐹‘𝑡)) = (𝐵 · (𝐹‘𝑥))) | 
| 31 | 20, 28, 30 | cbvmpt 5252 | . . . 4 ⊢ (𝑡 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑡))) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥))) | 
| 32 | 24, 31 | eqtrdi 2792 | . . 3 ⊢ (𝜑 → (𝑡 ∈ 𝐴 ↦ ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥))‘(𝐹‘𝑡))) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥)))) | 
| 33 | 11, 32 | eqtrd 2776 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥)))) | 
| 34 | 7, 4 | cncfco 24934 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) ∘ 𝐹) ∈ (𝐴–cn→ℂ)) | 
| 35 | 33, 34 | eqeltrrd 2841 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥))) ∈ (𝐴–cn→ℂ)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 Ⅎwnf 1782 ∈ wcel 2107 Ⅎwnfc 2889 ↦ cmpt 5224 ∘ ccom 5688 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ℂcc 11154 · cmul 11161 –cn→ccncf 24903 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-sup 9483 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-n0 12529 df-z 12616 df-uz 12880 df-rp 13036 df-seq 14044 df-exp 14104 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-cncf 24905 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |