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Mirrors > Home > MPE Home > Th. List > grpsubfvalALT | Structured version Visualization version GIF version |
Description: Shorter proof of grpsubfval 18786 using ax-rep 5240. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 19-Feb-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grpsubval.b | ⊢ 𝐵 = (Base‘𝐺) |
grpsubval.p | ⊢ + = (+g‘𝐺) |
grpsubval.i | ⊢ 𝐼 = (invg‘𝐺) |
grpsubval.m | ⊢ − = (-g‘𝐺) |
Ref | Expression |
---|---|
grpsubfvalALT | ⊢ − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpsubval.m | . . 3 ⊢ − = (-g‘𝐺) | |
2 | fveq2 6839 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
3 | grpsubval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
4 | 2, 3 | eqtr4di 2794 | . . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵) |
5 | fveq2 6839 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺)) | |
6 | grpsubval.p | . . . . . . 7 ⊢ + = (+g‘𝐺) | |
7 | 5, 6 | eqtr4di 2794 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + ) |
8 | eqidd 2737 | . . . . . 6 ⊢ (𝑔 = 𝐺 → 𝑥 = 𝑥) | |
9 | fveq2 6839 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (invg‘𝑔) = (invg‘𝐺)) | |
10 | grpsubval.i | . . . . . . . 8 ⊢ 𝐼 = (invg‘𝐺) | |
11 | 9, 10 | eqtr4di 2794 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (invg‘𝑔) = 𝐼) |
12 | 11 | fveq1d 6841 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ((invg‘𝑔)‘𝑦) = (𝐼‘𝑦)) |
13 | 7, 8, 12 | oveq123d 7374 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦)) = (𝑥 + (𝐼‘𝑦))) |
14 | 4, 4, 13 | mpoeq123dv 7428 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))) |
15 | df-sbg 18745 | . . . 4 ⊢ -g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦)))) | |
16 | 3 | fvexi 6853 | . . . . 5 ⊢ 𝐵 ∈ V |
17 | 16, 16 | mpoex 8008 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) ∈ V |
18 | 14, 15, 17 | fvmpt 6945 | . . 3 ⊢ (𝐺 ∈ V → (-g‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))) |
19 | 1, 18 | eqtrid 2788 | . 2 ⊢ (𝐺 ∈ V → − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))) |
20 | fvprc 6831 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (-g‘𝐺) = ∅) | |
21 | 1, 20 | eqtrid 2788 | . . 3 ⊢ (¬ 𝐺 ∈ V → − = ∅) |
22 | fvprc 6831 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅) | |
23 | 3, 22 | eqtrid 2788 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅) |
24 | 23 | olcd 872 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅)) |
25 | 0mpo0 7436 | . . . 4 ⊢ ((𝐵 = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) = ∅) | |
26 | 24, 25 | syl 17 | . . 3 ⊢ (¬ 𝐺 ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) = ∅) |
27 | 21, 26 | eqtr4d 2779 | . 2 ⊢ (¬ 𝐺 ∈ V → − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))) |
28 | 19, 27 | pm2.61i 182 | 1 ⊢ − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 845 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ∅c0 4280 ‘cfv 6493 (class class class)co 7353 ∈ cmpo 7355 Basecbs 17075 +gcplusg 17125 invgcminusg 18741 -gcsg 18742 |
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 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 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-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7356 df-oprab 7357 df-mpo 7358 df-1st 7917 df-2nd 7918 df-sbg 18745 |
This theorem is referenced by: (None) |
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