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Mirrors > Home > MPE Home > Th. List > grpsubfvalALT | Structured version Visualization version GIF version |
Description: Shorter proof of grpsubfval 18265 using ax-rep 5154. (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 6674 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
3 | grpsubval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
4 | 2, 3 | eqtr4di 2791 | . . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵) |
5 | fveq2 6674 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺)) | |
6 | grpsubval.p | . . . . . . 7 ⊢ + = (+g‘𝐺) | |
7 | 5, 6 | eqtr4di 2791 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + ) |
8 | eqidd 2739 | . . . . . 6 ⊢ (𝑔 = 𝐺 → 𝑥 = 𝑥) | |
9 | fveq2 6674 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (invg‘𝑔) = (invg‘𝐺)) | |
10 | grpsubval.i | . . . . . . . 8 ⊢ 𝐼 = (invg‘𝐺) | |
11 | 9, 10 | eqtr4di 2791 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (invg‘𝑔) = 𝐼) |
12 | 11 | fveq1d 6676 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ((invg‘𝑔)‘𝑦) = (𝐼‘𝑦)) |
13 | 7, 8, 12 | oveq123d 7191 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦)) = (𝑥 + (𝐼‘𝑦))) |
14 | 4, 4, 13 | mpoeq123dv 7243 | . . . 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 18224 | . . . 4 ⊢ -g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦)))) | |
16 | 3 | fvexi 6688 | . . . . 5 ⊢ 𝐵 ∈ V |
17 | 16, 16 | mpoex 7803 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) ∈ V |
18 | 14, 15, 17 | fvmpt 6775 | . . 3 ⊢ (𝐺 ∈ V → (-g‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))) |
19 | 1, 18 | syl5eq 2785 | . 2 ⊢ (𝐺 ∈ V → − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))) |
20 | fvprc 6666 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (-g‘𝐺) = ∅) | |
21 | 1, 20 | syl5eq 2785 | . . 3 ⊢ (¬ 𝐺 ∈ V → − = ∅) |
22 | fvprc 6666 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅) | |
23 | 3, 22 | syl5eq 2785 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅) |
24 | 23 | olcd 873 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅)) |
25 | 0mpo0 7251 | . . . 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 2776 | . 2 ⊢ (¬ 𝐺 ∈ V → − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))) |
28 | 19, 27 | pm2.61i 185 | 1 ⊢ − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 846 = wceq 1542 ∈ wcel 2114 Vcvv 3398 ∅c0 4211 ‘cfv 6339 (class class class)co 7170 ∈ cmpo 7172 Basecbs 16586 +gcplusg 16668 invgcminusg 18220 -gcsg 18221 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-ov 7173 df-oprab 7174 df-mpo 7175 df-1st 7714 df-2nd 7715 df-sbg 18224 |
This theorem is referenced by: (None) |
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