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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 6735 | . . 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 4183 | . 2 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
7 | fvconst2g 7156 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑋 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵) | |
8 | 1, 7 | sylan 581 | . 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 7633 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (((𝐴 × {𝐵}) ∘f 𝑅𝐹)‘𝑋) = (𝐵𝑅𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {csn 4591 × cxp 5636 Fn wfn 6496 ‘cfv 6501 (class class class)co 7362 ∘f cof 7620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 |
This theorem is referenced by: ofnegsub 12158 pwsvscaval 17384 lmhmvsca 20522 psrvscaval 21376 mplvscaval 21436 coe1sclmulfv 21670 mamuvs1 21768 mamuvs2 21769 matvscacell 21801 mdetrsca 21968 mbfmulc2lem 25027 i1fmulclem 25083 itg1mulc 25085 itg2monolem1 25131 uc1pmon1p 25532 coemulc 25632 basellem9 26454 ofdivrec 42680 |
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