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Mirrors > Home > MPE Home > Th. List > grpsubfvalALT | Structured version Visualization version GIF version |
Description: Shorter proof of grpsubfval 18142 using ax-rep 5183. (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 6663 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
3 | grpsubval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
4 | 2, 3 | syl6eqr 2873 | . . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵) |
5 | fveq2 6663 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺)) | |
6 | grpsubval.p | . . . . . . 7 ⊢ + = (+g‘𝐺) | |
7 | 5, 6 | syl6eqr 2873 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + ) |
8 | eqidd 2821 | . . . . . 6 ⊢ (𝑔 = 𝐺 → 𝑥 = 𝑥) | |
9 | fveq2 6663 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (invg‘𝑔) = (invg‘𝐺)) | |
10 | grpsubval.i | . . . . . . . 8 ⊢ 𝐼 = (invg‘𝐺) | |
11 | 9, 10 | syl6eqr 2873 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (invg‘𝑔) = 𝐼) |
12 | 11 | fveq1d 6665 | . . . . . 6 ⊢ (𝑔 = 𝐺 → ((invg‘𝑔)‘𝑦) = (𝐼‘𝑦)) |
13 | 7, 8, 12 | oveq123d 7170 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦)) = (𝑥 + (𝐼‘𝑦))) |
14 | 4, 4, 13 | mpoeq123dv 7222 | . . . 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 18103 | . . . 4 ⊢ -g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦)))) | |
16 | 3 | fvexi 6677 | . . . . 5 ⊢ 𝐵 ∈ V |
17 | 16, 16 | mpoex 7770 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) ∈ V |
18 | 14, 15, 17 | fvmpt 6761 | . . 3 ⊢ (𝐺 ∈ V → (-g‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))) |
19 | 1, 18 | syl5eq 2867 | . 2 ⊢ (𝐺 ∈ V → − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))) |
20 | fvprc 6656 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (-g‘𝐺) = ∅) | |
21 | 1, 20 | syl5eq 2867 | . . 3 ⊢ (¬ 𝐺 ∈ V → − = ∅) |
22 | fvprc 6656 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅) | |
23 | 3, 22 | syl5eq 2867 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅) |
24 | 23 | olcd 870 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅)) |
25 | 0mpo0 7230 | . . . 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 2858 | . 2 ⊢ (¬ 𝐺 ∈ V → − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))) |
28 | 19, 27 | pm2.61i 184 | 1 ⊢ − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 843 = wceq 1536 ∈ wcel 2113 Vcvv 3491 ∅c0 4284 ‘cfv 6348 (class class class)co 7149 ∈ cmpo 7151 Basecbs 16478 +gcplusg 16560 invgcminusg 18099 -gcsg 18100 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7152 df-oprab 7153 df-mpo 7154 df-1st 7682 df-2nd 7683 df-sbg 18103 |
This theorem is referenced by: (None) |
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