Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > scafeq | Structured version Visualization version GIF version |
Description: If the scalar multiplication operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
scaffval.b | ⊢ 𝐵 = (Base‘𝑊) |
scaffval.f | ⊢ 𝐹 = (Scalar‘𝑊) |
scaffval.k | ⊢ 𝐾 = (Base‘𝐹) |
scaffval.a | ⊢ ∙ = ( ·sf ‘𝑊) |
scaffval.s | ⊢ · = ( ·𝑠 ‘𝑊) |
Ref | Expression |
---|---|
scafeq | ⊢ ( · Fn (𝐾 × 𝐵) → ∙ = · ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnov 7282 | . . 3 ⊢ ( · Fn (𝐾 × 𝐵) ↔ · = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦))) | |
2 | 1 | biimpi 218 | . 2 ⊢ ( · Fn (𝐾 × 𝐵) → · = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦))) |
3 | scaffval.b | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
4 | scaffval.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
5 | scaffval.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
6 | scaffval.a | . . 3 ⊢ ∙ = ( ·sf ‘𝑊) | |
7 | scaffval.s | . . 3 ⊢ · = ( ·𝑠 ‘𝑊) | |
8 | 3, 4, 5, 6, 7 | scaffval 19652 | . 2 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)) |
9 | 2, 8 | syl6reqr 2875 | 1 ⊢ ( · Fn (𝐾 × 𝐵) → ∙ = · ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 × cxp 5553 Fn wfn 6350 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 Basecbs 16483 Scalarcsca 16568 ·𝑠 cvsca 16569 ·sf cscaf 19635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-scaf 19637 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |