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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 | fnov 7285 | . . 3 ⊢ ( + Fn (𝐵 × 𝐵) ↔ + = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))) | |
2 | 1 | biimpi 218 | . 2 ⊢ ( + Fn (𝐵 × 𝐵) → + = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))) |
3 | plusffval.1 | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
4 | plusffval.2 | . . 3 ⊢ + = (+g‘𝐺) | |
5 | plusffval.3 | . . 3 ⊢ ⨣ = (+𝑓‘𝐺) | |
6 | 3, 4, 5 | plusffval 17861 | . 2 ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) |
7 | 2, 6 | syl6reqr 2878 | 1 ⊢ ( + Fn (𝐵 × 𝐵) → ⨣ = + ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 × cxp 5556 Fn wfn 6353 ‘cfv 6358 (class class class)co 7159 ∈ cmpo 7161 Basecbs 16486 +gcplusg 16568 +𝑓cplusf 17852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-plusf 17854 |
This theorem is referenced by: mgmb1mgm1 17868 mndfo 17938 cnfldplusf 20575 efmndtmd 22712 |
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