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Mirrors > Home > MPE Home > Th. List > grpinvfvalALT | Structured version Visualization version GIF version |
Description: Shorter proof of grpinvfval 18794 using ax-rep 5243. (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 6843 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
3 | grpinvval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
4 | 2, 3 | eqtr4di 2791 | . . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵) |
5 | fveq2 6843 | . . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺)) | |
6 | grpinvval.p | . . . . . . . . 9 ⊢ + = (+g‘𝐺) | |
7 | 5, 6 | eqtr4di 2791 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + ) |
8 | 7 | oveqd 7375 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑦(+g‘𝑔)𝑥) = (𝑦 + 𝑥)) |
9 | fveq2 6843 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺)) | |
10 | grpinvval.o | . . . . . . . 8 ⊢ 0 = (0g‘𝐺) | |
11 | 9, 10 | eqtr4di 2791 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 ) |
12 | 8, 11 | eqeq12d 2749 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑦(+g‘𝑔)𝑥) = (0g‘𝑔) ↔ (𝑦 + 𝑥) = 0 )) |
13 | 4, 12 | riotaeqbidv 7317 | . . . . 5 ⊢ (𝑔 = 𝐺 → (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔)) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) |
14 | 4, 13 | mpteq12dv 5197 | . . . 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 18757 | . . . 4 ⊢ invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔)))) | |
16 | 14, 15, 3 | mptfvmpt 7179 | . . 3 ⊢ (𝐺 ∈ V → (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))) |
17 | fvprc 6835 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → (invg‘𝐺) = ∅) | |
18 | mpt0 6644 | . . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) = ∅ | |
19 | 17, 18 | eqtr4di 2791 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (invg‘𝐺) = (𝑥 ∈ ∅ ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))) |
20 | fvprc 6835 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅) | |
21 | 3, 20 | eqtrid 2785 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅) |
22 | 21 | mpteq1d 5201 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) = (𝑥 ∈ ∅ ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))) |
23 | 19, 22 | eqtr4d 2776 | . . 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 2761 | 1 ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ∅c0 4283 ↦ cmpt 5189 ‘cfv 6497 ℩crio 7313 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 0gc0g 17326 invgcminusg 18754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-minusg 18757 |
This theorem is referenced by: (None) |
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