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| Mirrors > Home > MPE Home > Th. List > plusffval | Structured version Visualization version GIF version | ||
| Description: The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof shortened by AV, 2-Mar-2024.) |
| Ref | Expression |
|---|---|
| plusffval.1 | ⊢ 𝐵 = (Base‘𝐺) |
| plusffval.2 | ⊢ + = (+g‘𝐺) |
| plusffval.3 | ⊢ ⨣ = (+𝑓‘𝐺) |
| Ref | Expression |
|---|---|
| plusffval | ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | plusffval.3 | . 2 ⊢ ⨣ = (+𝑓‘𝐺) | |
| 2 | fveq2 6906 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
| 3 | plusffval.1 | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
| 4 | 2, 3 | eqtr4di 2795 | . . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵) |
| 5 | fveq2 6906 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺)) | |
| 6 | plusffval.2 | . . . . . . 7 ⊢ + = (+g‘𝐺) | |
| 7 | 5, 6 | eqtr4di 2795 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + ) |
| 8 | 7 | oveqd 7448 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)𝑦) = (𝑥 + 𝑦)) |
| 9 | 4, 4, 8 | mpoeq123dv 7508 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))) |
| 10 | df-plusf 18652 | . . . 4 ⊢ +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦))) | |
| 11 | 3 | fvexi 6920 | . . . . 5 ⊢ 𝐵 ∈ V |
| 12 | 6 | fvexi 6920 | . . . . . . 7 ⊢ + ∈ V |
| 13 | 12 | rnex 7932 | . . . . . 6 ⊢ ran + ∈ V |
| 14 | p0ex 5384 | . . . . . 6 ⊢ {∅} ∈ V | |
| 15 | 13, 14 | unex 7764 | . . . . 5 ⊢ (ran + ∪ {∅}) ∈ V |
| 16 | df-ov 7434 | . . . . . . 7 ⊢ (𝑥 + 𝑦) = ( + ‘〈𝑥, 𝑦〉) | |
| 17 | fvrn0 6936 | . . . . . . 7 ⊢ ( + ‘〈𝑥, 𝑦〉) ∈ (ran + ∪ {∅}) | |
| 18 | 16, 17 | eqeltri 2837 | . . . . . 6 ⊢ (𝑥 + 𝑦) ∈ (ran + ∪ {∅}) |
| 19 | 18 | rgen2w 3066 | . . . . 5 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ∈ (ran + ∪ {∅}) |
| 20 | 11, 11, 15, 19 | mpoexw 8103 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) ∈ V |
| 21 | 9, 10, 20 | fvmpt 7016 | . . 3 ⊢ (𝐺 ∈ V → (+𝑓‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))) |
| 22 | fvprc 6898 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (+𝑓‘𝐺) = ∅) | |
| 23 | fvprc 6898 | . . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅) | |
| 24 | 3, 23 | eqtrid 2789 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅) |
| 25 | 24 | olcd 875 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅)) |
| 26 | 0mpo0 7516 | . . . . 5 ⊢ ((𝐵 = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) = ∅) | |
| 27 | 25, 26 | syl 17 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) = ∅) |
| 28 | 22, 27 | eqtr4d 2780 | . . 3 ⊢ (¬ 𝐺 ∈ V → (+𝑓‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))) |
| 29 | 21, 28 | pm2.61i 182 | . 2 ⊢ (+𝑓‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) |
| 30 | 1, 29 | eqtri 2765 | 1 ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∨ wo 848 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ∅c0 4333 {csn 4626 〈cop 4632 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 Basecbs 17247 +gcplusg 17297 +𝑓cplusf 18650 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-plusf 18652 |
| This theorem is referenced by: plusfval 18660 plusfeq 18661 plusffn 18662 mgmplusf 18663 rlmscaf 21214 istgp2 24099 oppgtmd 24105 submtmd 24112 prdstmdd 24132 ressplusf 32948 pl1cn 33954 |
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