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Mirrors > Home > MPE Home > Th. List > ofc12 | Structured version Visualization version GIF version |
Description: Function operation on two constant functions. (Contributed by Mario Carneiro, 28-Jul-2014.) |
Ref | Expression |
---|---|
ofc12.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
ofc12.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
ofc12.3 | ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
Ref | Expression |
---|---|
ofc12 | ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅(𝐴 × {𝐶})) = (𝐴 × {(𝐵𝑅𝐶)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ofc12.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | ofc12.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
3 | 2 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) |
4 | ofc12.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
5 | 4 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑋) |
6 | fconstmpt 5608 | . . . 4 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)) |
8 | fconstmpt 5608 | . . . 4 ⊢ (𝐴 × {𝐶}) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴 × {𝐶}) = (𝑥 ∈ 𝐴 ↦ 𝐶)) |
10 | 1, 3, 5, 7, 9 | offval2 7420 | . 2 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅(𝐴 × {𝐶})) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))) |
11 | fconstmpt 5608 | . 2 ⊢ (𝐴 × {(𝐵𝑅𝐶)}) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)) | |
12 | 10, 11 | syl6eqr 2874 | 1 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅(𝐴 × {𝐶})) = (𝐴 × {(𝐵𝑅𝐶)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {csn 4560 ↦ cmpt 5138 × cxp 5547 (class class class)co 7150 ∘f cof 7401 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 |
This theorem is referenced by: pwsdiagmhm 17989 pwsdiaglmhm 19823 psrlmod 20175 coe1mul2 20431 itg2mulc 24342 dgrmulc 24855 lflvsdi2a 36210 lflvsass 36211 lflsc0N 36213 mendlmod 39786 expgrowth 40660 |
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