Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6560 | . . 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 4192 | . 2 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
7 | fvconst2g 6956 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑋 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵) | |
8 | 1, 7 | sylan 580 | . 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 7407 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (((𝐴 × {𝐵}) ∘f 𝑅𝐹)‘𝑋) = (𝐵𝑅𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {csn 4557 × cxp 5546 Fn wfn 6343 ‘cfv 6348 (class class class)co 7145 ∘f cof 7396 |
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-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 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-ral 3140 df-rex 3141 df-reu 3142 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-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 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-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 |
This theorem is referenced by: ofnegsub 11624 pwsvscaval 16756 lmhmvsca 19746 psrvscaval 20100 mplvscaval 20156 coe1sclmulfv 20379 mamuvs1 20942 mamuvs2 20943 matvscacell 20973 mdetrsca 21140 mbfmulc2lem 24175 i1fmulclem 24230 itg1mulc 24232 itg2monolem1 24278 uc1pmon1p 24672 coemulc 24772 basellem9 25593 ofdivrec 40535 |
Copyright terms: Public domain | W3C validator |