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Mirrors > Home > MPE Home > Th. List > ofc1 | Structured version Visualization version GIF version |
Description: Left operation by a constant. (Contributed by Mario Carneiro, 24-Jul-2014.) |
Ref | Expression |
---|---|
ofc1.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
ofc1.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
ofc1.3 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
ofc1.4 | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶) |
Ref | Expression |
---|---|
ofc1 | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (((𝐴 × {𝐵}) ∘f 𝑅𝐹)‘𝑋) = (𝐵𝑅𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ofc1.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
2 | fnconstg 6541 | . . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐴 × {𝐵}) Fn 𝐴) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 × {𝐵}) Fn 𝐴) |
4 | ofc1.3 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
5 | ofc1.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
6 | inidm 4145 | . 2 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
7 | fvconst2g 6941 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑋 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵) | |
8 | 1, 7 | sylan 583 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵) |
9 | ofc1.4 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶) | |
10 | 3, 4, 5, 5, 6, 8, 9 | ofval 7398 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (((𝐴 × {𝐵}) ∘f 𝑅𝐹)‘𝑋) = (𝐵𝑅𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {csn 4525 × cxp 5517 Fn wfn 6319 ‘cfv 6324 (class class class)co 7135 ∘f cof 7387 |
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-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 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-ral 3111 df-rex 3112 df-reu 3113 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-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 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-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 |
This theorem is referenced by: ofnegsub 11623 pwsvscaval 16760 lmhmvsca 19810 psrvscaval 20630 mplvscaval 20687 coe1sclmulfv 20912 mamuvs1 21010 mamuvs2 21011 matvscacell 21041 mdetrsca 21208 mbfmulc2lem 24251 i1fmulclem 24306 itg1mulc 24308 itg2monolem1 24354 uc1pmon1p 24752 coemulc 24852 basellem9 25674 ofdivrec 41030 |
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