![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > plusfeq | Structured version Visualization version GIF version |
Description: If the addition operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
plusffval.1 | ⊢ 𝐵 = (Base‘𝐺) |
plusffval.2 | ⊢ + = (+g‘𝐺) |
plusffval.3 | ⊢ ⨣ = (+𝑓‘𝐺) |
Ref | Expression |
---|---|
plusfeq | ⊢ ( + Fn (𝐵 × 𝐵) → ⨣ = + ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | plusffval.1 | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | plusffval.2 | . . 3 ⊢ + = (+g‘𝐺) | |
3 | plusffval.3 | . . 3 ⊢ ⨣ = (+𝑓‘𝐺) | |
4 | 1, 2, 3 | plusffval 18463 | . 2 ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) |
5 | fnov 7481 | . . 3 ⊢ ( + Fn (𝐵 × 𝐵) ↔ + = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))) | |
6 | 5 | biimpi 215 | . 2 ⊢ ( + Fn (𝐵 × 𝐵) → + = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))) |
7 | 4, 6 | eqtr4id 2796 | 1 ⊢ ( + Fn (𝐵 × 𝐵) → ⨣ = + ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 × cxp 5629 Fn wfn 6488 ‘cfv 6493 (class class class)co 7351 ∈ cmpo 7353 Basecbs 17043 +gcplusg 17093 +𝑓cplusf 18454 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-1st 7913 df-2nd 7914 df-plusf 18456 |
This theorem is referenced by: mgmb1mgm1 18470 mndfo 18540 cnfldplusf 20777 efmndtmd 23404 |
Copyright terms: Public domain | W3C validator |