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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 6839 | . . . . . 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 6839 | . . . . . . 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 7368 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)𝑦) = (𝑥 + 𝑦)) |
9 | 4, 4, 8 | mpoeq123dv 7426 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))) |
10 | df-plusf 18456 | . . . 4 ⊢ +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦))) | |
11 | 3 | fvexi 6853 | . . . . 5 ⊢ 𝐵 ∈ V |
12 | 6 | fvexi 6853 | . . . . . . 7 ⊢ + ∈ V |
13 | 12 | rnex 7841 | . . . . . 6 ⊢ ran + ∈ V |
14 | p0ex 5337 | . . . . . 6 ⊢ {∅} ∈ V | |
15 | 13, 14 | unex 7672 | . . . . 5 ⊢ (ran + ∪ {∅}) ∈ V |
16 | df-ov 7354 | . . . . . . 7 ⊢ (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩) | |
17 | fvrn0 6869 | . . . . . . 7 ⊢ ( + ‘⟨𝑥, 𝑦⟩) ∈ (ran + ∪ {∅}) | |
18 | 16, 17 | eqeltri 2834 | . . . . . 6 ⊢ (𝑥 + 𝑦) ∈ (ran + ∪ {∅}) |
19 | 18 | rgen2w 3067 | . . . . 5 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ∈ (ran + ∪ {∅}) |
20 | 11, 11, 15, 19 | mpoexw 8003 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) ∈ V |
21 | 9, 10, 20 | fvmpt 6945 | . . 3 ⊢ (𝐺 ∈ V → (+𝑓‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))) |
22 | fvprc 6831 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (+𝑓‘𝐺) = ∅) | |
23 | fvprc 6831 | . . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅) | |
24 | 3, 23 | eqtrid 2789 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅) |
25 | 24 | olcd 872 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅)) |
26 | 0mpo0 7434 | . . . . 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 845 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ∪ cun 3906 ∅c0 4280 {csn 4584 ⟨cop 4590 ran crn 5632 ‘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: plusfval 18464 plusfeq 18465 plusffn 18466 mgmplusf 18467 rlmscaf 20631 istgp2 23394 oppgtmd 23400 submtmd 23407 prdstmdd 23427 ressplusf 31643 pl1cn 32348 |
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