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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 24945 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 2735 | . . . . . . 7 ⊢ (𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) = (𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) | |
3 | 2 | mulc1cncf 24945 | . . . . . 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 24933 | . . . . 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 24933 | . . . . 5 ⊢ (𝐹 ∈ (𝐴–cn→ℂ) → 𝐹:𝐴⟶ℂ) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
10 | fcompt 7153 | . . . 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 7104 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → (𝐹‘𝑡) ∈ ℂ) |
13 | 1 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → 𝐵 ∈ ℂ) |
14 | 13, 12 | mulcld 11279 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → (𝐵 · (𝐹‘𝑡)) ∈ ℂ) |
15 | mulc1cncfg.1 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐹 | |
16 | nfcv 2903 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑡 | |
17 | 15, 16 | nffv 6917 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑡) |
18 | nfcv 2903 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐵 | |
19 | nfcv 2903 | . . . . . . . 8 ⊢ Ⅎ𝑥 · | |
20 | 18, 19, 17 | nfov 7461 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐵 · (𝐹‘𝑡)) |
21 | oveq2 7439 | . . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑡) → (𝐵 · 𝑥) = (𝐵 · (𝐹‘𝑡))) | |
22 | 17, 20, 21, 2 | fvmptf 7037 | . . . . . 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 5248 | . . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝐴 ↦ ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥))‘(𝐹‘𝑡))) = (𝑡 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑡)))) |
25 | nfcv 2903 | . . . . . 6 ⊢ Ⅎ𝑡𝐵 | |
26 | nfcv 2903 | . . . . . 6 ⊢ Ⅎ𝑡 · | |
27 | nfcv 2903 | . . . . . 6 ⊢ Ⅎ𝑡(𝐹‘𝑥) | |
28 | 25, 26, 27 | nfov 7461 | . . . . 5 ⊢ Ⅎ𝑡(𝐵 · (𝐹‘𝑥)) |
29 | fveq2 6907 | . . . . . 6 ⊢ (𝑡 = 𝑥 → (𝐹‘𝑡) = (𝐹‘𝑥)) | |
30 | 29 | oveq2d 7447 | . . . . 5 ⊢ (𝑡 = 𝑥 → (𝐵 · (𝐹‘𝑡)) = (𝐵 · (𝐹‘𝑥))) |
31 | 20, 28, 30 | cbvmpt 5259 | . . . 4 ⊢ (𝑡 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑡))) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥))) |
32 | 24, 31 | eqtrdi 2791 | . . 3 ⊢ (𝜑 → (𝑡 ∈ 𝐴 ↦ ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥))‘(𝐹‘𝑡))) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥)))) |
33 | 11, 32 | eqtrd 2775 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥)))) |
34 | 7, 4 | cncfco 24947 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝐵 · 𝑥)) ∘ 𝐹) ∈ (𝐴–cn→ℂ)) |
35 | 33, 34 | eqeltrrd 2840 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥))) ∈ (𝐴–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 Ⅎwnfc 2888 ↦ cmpt 5231 ∘ ccom 5693 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 · cmul 11158 –cn→ccncf 24916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-cncf 24918 |
This theorem is referenced by: (None) |
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