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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 6907 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
3 | plusffval.1 | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
4 | 2, 3 | eqtr4di 2793 | . . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵) |
5 | fveq2 6907 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺)) | |
6 | plusffval.2 | . . . . . . 7 ⊢ + = (+g‘𝐺) | |
7 | 5, 6 | eqtr4di 2793 | . . . . . 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 18665 | . . . 4 ⊢ +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦))) | |
11 | 3 | fvexi 6921 | . . . . 5 ⊢ 𝐵 ∈ V |
12 | 6 | fvexi 6921 | . . . . . . 7 ⊢ + ∈ V |
13 | 12 | rnex 7933 | . . . . . 6 ⊢ ran + ∈ V |
14 | p0ex 5390 | . . . . . 6 ⊢ {∅} ∈ V | |
15 | 13, 14 | unex 7763 | . . . . 5 ⊢ (ran + ∪ {∅}) ∈ V |
16 | df-ov 7434 | . . . . . . 7 ⊢ (𝑥 + 𝑦) = ( + ‘〈𝑥, 𝑦〉) | |
17 | fvrn0 6937 | . . . . . . 7 ⊢ ( + ‘〈𝑥, 𝑦〉) ∈ (ran + ∪ {∅}) | |
18 | 16, 17 | eqeltri 2835 | . . . . . 6 ⊢ (𝑥 + 𝑦) ∈ (ran + ∪ {∅}) |
19 | 18 | rgen2w 3064 | . . . . 5 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ∈ (ran + ∪ {∅}) |
20 | 11, 11, 15, 19 | mpoexw 8102 | . . . 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 6899 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (+𝑓‘𝐺) = ∅) | |
23 | fvprc 6899 | . . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅) | |
24 | 3, 23 | eqtrid 2787 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅) |
25 | 24 | olcd 874 | . . . . 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 2778 | . . 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 2763 | 1 ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 847 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∪ cun 3961 ∅c0 4339 {csn 4631 〈cop 4637 ran crn 5690 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 Basecbs 17245 +gcplusg 17298 +𝑓cplusf 18663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-plusf 18665 |
This theorem is referenced by: plusfval 18673 plusfeq 18674 plusffn 18675 mgmplusf 18676 rlmscaf 21232 istgp2 24115 oppgtmd 24121 submtmd 24128 prdstmdd 24148 ressplusf 32933 pl1cn 33916 |
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