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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 17850 | . 2 ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) |
5 | fnov 7261 | . . 3 ⊢ ( + Fn (𝐵 × 𝐵) ↔ + = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))) | |
6 | 5 | biimpi 219 | . 2 ⊢ ( + Fn (𝐵 × 𝐵) → + = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))) |
7 | 4, 6 | eqtr4id 2852 | 1 ⊢ ( + Fn (𝐵 × 𝐵) → ⨣ = + ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 × cxp 5517 Fn wfn 6319 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 Basecbs 16475 +gcplusg 16557 +𝑓cplusf 17841 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-plusf 17843 |
This theorem is referenced by: mgmb1mgm1 17857 mndfo 17927 cnfldplusf 20118 efmndtmd 22706 |
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