![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6878 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
3 | plusffval.1 | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
4 | 2, 3 | eqtr4di 2789 | . . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵) |
5 | fveq2 6878 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺)) | |
6 | plusffval.2 | . . . . . . 7 ⊢ + = (+g‘𝐺) | |
7 | 5, 6 | eqtr4di 2789 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + ) |
8 | 7 | oveqd 7410 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)𝑦) = (𝑥 + 𝑦)) |
9 | 4, 4, 8 | mpoeq123dv 7468 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))) |
10 | df-plusf 18542 | . . . 4 ⊢ +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦))) | |
11 | 3 | fvexi 6892 | . . . . 5 ⊢ 𝐵 ∈ V |
12 | 6 | fvexi 6892 | . . . . . . 7 ⊢ + ∈ V |
13 | 12 | rnex 7885 | . . . . . 6 ⊢ ran + ∈ V |
14 | p0ex 5375 | . . . . . 6 ⊢ {∅} ∈ V | |
15 | 13, 14 | unex 7716 | . . . . 5 ⊢ (ran + ∪ {∅}) ∈ V |
16 | df-ov 7396 | . . . . . . 7 ⊢ (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩) | |
17 | fvrn0 6908 | . . . . . . 7 ⊢ ( + ‘⟨𝑥, 𝑦⟩) ∈ (ran + ∪ {∅}) | |
18 | 16, 17 | eqeltri 2828 | . . . . . 6 ⊢ (𝑥 + 𝑦) ∈ (ran + ∪ {∅}) |
19 | 18 | rgen2w 3065 | . . . . 5 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ∈ (ran + ∪ {∅}) |
20 | 11, 11, 15, 19 | mpoexw 8047 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) ∈ V |
21 | 9, 10, 20 | fvmpt 6984 | . . 3 ⊢ (𝐺 ∈ V → (+𝑓‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))) |
22 | fvprc 6870 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (+𝑓‘𝐺) = ∅) | |
23 | fvprc 6870 | . . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅) | |
24 | 3, 23 | eqtrid 2783 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅) |
25 | 24 | olcd 872 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅)) |
26 | 0mpo0 7476 | . . . . 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 2774 | . . 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 2759 | 1 ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 845 = wceq 1541 ∈ wcel 2106 Vcvv 3473 ∪ cun 3942 ∅c0 4318 {csn 4622 ⟨cop 4628 ran crn 5670 ‘cfv 6532 (class class class)co 7393 ∈ cmpo 7395 Basecbs 17126 +gcplusg 17179 +𝑓cplusf 18540 |
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 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-fv 6540 df-ov 7396 df-oprab 7397 df-mpo 7398 df-1st 7957 df-2nd 7958 df-plusf 18542 |
This theorem is referenced by: plusfval 18550 plusfeq 18551 plusffn 18552 mgmplusf 18553 rlmscaf 20779 istgp2 23524 oppgtmd 23530 submtmd 23537 prdstmdd 23557 ressplusf 31998 pl1cn 32766 |
Copyright terms: Public domain | W3C validator |