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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 6663 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
3 | plusffval.1 | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
4 | 2, 3 | syl6eqr 2871 | . . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵) |
5 | fveq2 6663 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺)) | |
6 | plusffval.2 | . . . . . . 7 ⊢ + = (+g‘𝐺) | |
7 | 5, 6 | syl6eqr 2871 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + ) |
8 | 7 | oveqd 7162 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)𝑦) = (𝑥 + 𝑦)) |
9 | 4, 4, 8 | mpoeq123dv 7218 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))) |
10 | df-plusf 17839 | . . . 4 ⊢ +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦))) | |
11 | 3 | fvexi 6677 | . . . . 5 ⊢ 𝐵 ∈ V |
12 | 6 | fvexi 6677 | . . . . . . 7 ⊢ + ∈ V |
13 | 12 | rnex 7606 | . . . . . 6 ⊢ ran + ∈ V |
14 | p0ex 5275 | . . . . . 6 ⊢ {∅} ∈ V | |
15 | 13, 14 | unex 7458 | . . . . 5 ⊢ (ran + ∪ {∅}) ∈ V |
16 | df-ov 7148 | . . . . . . 7 ⊢ (𝑥 + 𝑦) = ( + ‘〈𝑥, 𝑦〉) | |
17 | fvrn0 6691 | . . . . . . 7 ⊢ ( + ‘〈𝑥, 𝑦〉) ∈ (ran + ∪ {∅}) | |
18 | 16, 17 | eqeltri 2906 | . . . . . 6 ⊢ (𝑥 + 𝑦) ∈ (ran + ∪ {∅}) |
19 | 18 | rgen2w 3148 | . . . . 5 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ∈ (ran + ∪ {∅}) |
20 | 11, 11, 15, 19 | mpoexw 7765 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) ∈ V |
21 | 9, 10, 20 | fvmpt 6761 | . . 3 ⊢ (𝐺 ∈ V → (+𝑓‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))) |
22 | fvprc 6656 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (+𝑓‘𝐺) = ∅) | |
23 | fvprc 6656 | . . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅) | |
24 | 3, 23 | syl5eq 2865 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅) |
25 | 24 | olcd 870 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅)) |
26 | 0mpo0 7226 | . . . . 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 2856 | . . 3 ⊢ (¬ 𝐺 ∈ V → (+𝑓‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))) |
29 | 21, 28 | pm2.61i 183 | . 2 ⊢ (+𝑓‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) |
30 | 1, 29 | eqtri 2841 | 1 ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 841 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∪ cun 3931 ∅c0 4288 {csn 4557 〈cop 4563 ran crn 5549 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 Basecbs 16471 +gcplusg 16553 +𝑓cplusf 17837 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-plusf 17839 |
This theorem is referenced by: plusfval 17847 plusfeq 17848 plusffn 17849 mgmplusf 17850 rlmscaf 19909 istgp2 22627 oppgtmd 22633 submtmd 22640 prdstmdd 22659 ressplusf 30564 pl1cn 31097 |
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