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| Mirrors > Home > MPE Home > Th. List > grpinvfvalALT | Structured version Visualization version GIF version | ||
| Description: Shorter proof of grpinvfval 18996 using ax-rep 5279. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| grpinvval.b | ⊢ 𝐵 = (Base‘𝐺) |
| grpinvval.p | ⊢ + = (+g‘𝐺) |
| grpinvval.o | ⊢ 0 = (0g‘𝐺) |
| grpinvval.n | ⊢ 𝑁 = (invg‘𝐺) |
| Ref | Expression |
|---|---|
| grpinvfvalALT | ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | grpinvval.n | . 2 ⊢ 𝑁 = (invg‘𝐺) | |
| 2 | fveq2 6906 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
| 3 | grpinvval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
| 4 | 2, 3 | eqtr4di 2795 | . . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵) |
| 5 | fveq2 6906 | . . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺)) | |
| 6 | grpinvval.p | . . . . . . . . 9 ⊢ + = (+g‘𝐺) | |
| 7 | 5, 6 | eqtr4di 2795 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + ) |
| 8 | 7 | oveqd 7448 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑦(+g‘𝑔)𝑥) = (𝑦 + 𝑥)) |
| 9 | fveq2 6906 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺)) | |
| 10 | grpinvval.o | . . . . . . . 8 ⊢ 0 = (0g‘𝐺) | |
| 11 | 9, 10 | eqtr4di 2795 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 ) |
| 12 | 8, 11 | eqeq12d 2753 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑦(+g‘𝑔)𝑥) = (0g‘𝑔) ↔ (𝑦 + 𝑥) = 0 )) |
| 13 | 4, 12 | riotaeqbidv 7391 | . . . . 5 ⊢ (𝑔 = 𝐺 → (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔)) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) |
| 14 | 4, 13 | mpteq12dv 5233 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔))) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))) |
| 15 | df-minusg 18955 | . . . 4 ⊢ invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔)))) | |
| 16 | 14, 15, 3 | mptfvmpt 7248 | . . 3 ⊢ (𝐺 ∈ V → (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))) |
| 17 | fvprc 6898 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → (invg‘𝐺) = ∅) | |
| 18 | mpt0 6710 | . . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) = ∅ | |
| 19 | 17, 18 | eqtr4di 2795 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (invg‘𝐺) = (𝑥 ∈ ∅ ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))) |
| 20 | fvprc 6898 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅) | |
| 21 | 3, 20 | eqtrid 2789 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅) |
| 22 | 21 | mpteq1d 5237 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) = (𝑥 ∈ ∅ ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))) |
| 23 | 19, 22 | eqtr4d 2780 | . . 3 ⊢ (¬ 𝐺 ∈ V → (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))) |
| 24 | 16, 23 | pm2.61i 182 | . 2 ⊢ (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) |
| 25 | 1, 24 | eqtri 2765 | 1 ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 ↦ cmpt 5225 ‘cfv 6561 ℩crio 7387 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 0gc0g 17484 invgcminusg 18952 |
| 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-pr 5432 |
| 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-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-riota 7388 df-ov 7434 df-minusg 18955 |
| This theorem is referenced by: (None) |
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