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Mirrors > Home > MPE Home > Th. List > grpinvfvalALT | Structured version Visualization version GIF version |
Description: Shorter proof of grpinvfval 18926 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 6891 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
3 | grpinvval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
4 | 2, 3 | eqtr4di 2785 | . . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵) |
5 | fveq2 6891 | . . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺)) | |
6 | grpinvval.p | . . . . . . . . 9 ⊢ + = (+g‘𝐺) | |
7 | 5, 6 | eqtr4di 2785 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + ) |
8 | 7 | oveqd 7431 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑦(+g‘𝑔)𝑥) = (𝑦 + 𝑥)) |
9 | fveq2 6891 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺)) | |
10 | grpinvval.o | . . . . . . . 8 ⊢ 0 = (0g‘𝐺) | |
11 | 9, 10 | eqtr4di 2785 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 ) |
12 | 8, 11 | eqeq12d 2743 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑦(+g‘𝑔)𝑥) = (0g‘𝑔) ↔ (𝑦 + 𝑥) = 0 )) |
13 | 4, 12 | riotaeqbidv 7373 | . . . . 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 18885 | . . . 4 ⊢ invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑦 ∈ (Base‘𝑔)(𝑦(+g‘𝑔)𝑥) = (0g‘𝑔)))) | |
16 | 14, 15, 3 | mptfvmpt 7234 | . . 3 ⊢ (𝐺 ∈ V → (invg‘𝐺) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))) |
17 | fvprc 6883 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → (invg‘𝐺) = ∅) | |
18 | mpt0 6691 | . . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) = ∅ | |
19 | 17, 18 | eqtr4di 2785 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (invg‘𝐺) = (𝑥 ∈ ∅ ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))) |
20 | fvprc 6883 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅) | |
21 | 3, 20 | eqtrid 2779 | . . . . 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 2770 | . . 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 2755 | 1 ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1534 ∈ wcel 2099 Vcvv 3469 ∅c0 4318 ↦ cmpt 5225 ‘cfv 6542 ℩crio 7369 (class class class)co 7414 Basecbs 17171 +gcplusg 17224 0gc0g 17412 invgcminusg 18882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-minusg 18885 |
This theorem is referenced by: (None) |
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