| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > grpsubfvalALT | Structured version Visualization version GIF version | ||
| Description: Shorter proof of grpsubfval 19001 using ax-rep 5279. (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 6906 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
| 3 | grpsubval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
| 4 | 2, 3 | eqtr4di 2795 | . . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵) |
| 5 | fveq2 6906 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺)) | |
| 6 | grpsubval.p | . . . . . . 7 ⊢ + = (+g‘𝐺) | |
| 7 | 5, 6 | eqtr4di 2795 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + ) |
| 8 | eqidd 2738 | . . . . . 6 ⊢ (𝑔 = 𝐺 → 𝑥 = 𝑥) | |
| 9 | fveq2 6906 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (invg‘𝑔) = (invg‘𝐺)) | |
| 10 | grpsubval.i | . . . . . . . 8 ⊢ 𝐼 = (invg‘𝐺) | |
| 11 | 9, 10 | eqtr4di 2795 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (invg‘𝑔) = 𝐼) |
| 12 | 11 | fveq1d 6908 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ((invg‘𝑔)‘𝑦) = (𝐼‘𝑦)) |
| 13 | 7, 8, 12 | oveq123d 7452 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦)) = (𝑥 + (𝐼‘𝑦))) |
| 14 | 4, 4, 13 | mpoeq123dv 7508 | . . . 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 18956 | . . . 4 ⊢ -g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦)))) | |
| 16 | 3 | fvexi 6920 | . . . . 5 ⊢ 𝐵 ∈ V |
| 17 | 16, 16 | mpoex 8104 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) ∈ V |
| 18 | 14, 15, 17 | fvmpt 7016 | . . 3 ⊢ (𝐺 ∈ V → (-g‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))) |
| 19 | 1, 18 | eqtrid 2789 | . 2 ⊢ (𝐺 ∈ V → − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))) |
| 20 | fvprc 6898 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (-g‘𝐺) = ∅) | |
| 21 | 1, 20 | eqtrid 2789 | . . 3 ⊢ (¬ 𝐺 ∈ V → − = ∅) |
| 22 | fvprc 6898 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅) | |
| 23 | 3, 22 | eqtrid 2789 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅) |
| 24 | 23 | olcd 875 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅)) |
| 25 | 0mpo0 7516 | . . . 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 2780 | . 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 848 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 Basecbs 17247 +gcplusg 17297 invgcminusg 18952 -gcsg 18953 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-sbg 18956 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |