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Mirrors > Home > MPE Home > Th. List > grpidd | Structured version Visualization version GIF version |
Description: Deduce the identity element of a magma from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.) |
Ref | Expression |
---|---|
grpidd.b | ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) |
grpidd.p | ⊢ (𝜑 → + = (+g‘𝐺)) |
grpidd.z | ⊢ (𝜑 → 0 ∈ 𝐵) |
grpidd.i | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) |
grpidd.j | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥) |
Ref | Expression |
---|---|
grpidd | ⊢ (𝜑 → 0 = (0g‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . 2 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2736 | . 2 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
3 | eqid 2736 | . 2 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | grpidd.z | . . 3 ⊢ (𝜑 → 0 ∈ 𝐵) | |
5 | grpidd.b | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) | |
6 | 4, 5 | eleqtrd 2840 | . 2 ⊢ (𝜑 → 0 ∈ (Base‘𝐺)) |
7 | 5 | eleq2d 2823 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐺))) |
8 | 7 | biimpar 478 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ 𝐵) |
9 | grpidd.p | . . . . . 6 ⊢ (𝜑 → + = (+g‘𝐺)) | |
10 | 9 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → + = (+g‘𝐺)) |
11 | 10 | oveqd 7373 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = ( 0 (+g‘𝐺)𝑥)) |
12 | grpidd.i | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) | |
13 | 11, 12 | eqtr3d 2778 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 (+g‘𝐺)𝑥) = 𝑥) |
14 | 8, 13 | syldan 591 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺)𝑥) = 𝑥) |
15 | 10 | oveqd 7373 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = (𝑥(+g‘𝐺) 0 )) |
16 | grpidd.j | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥) | |
17 | 15, 16 | eqtr3d 2778 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝐺) 0 ) = 𝑥) |
18 | 8, 17 | syldan 591 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺) 0 ) = 𝑥) |
19 | 1, 2, 3, 6, 14, 18 | ismgmid2 18522 | 1 ⊢ (𝜑 → 0 = (0g‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6496 (class class class)co 7356 Basecbs 17082 +gcplusg 17132 0gc0g 17320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-iota 6448 df-fun 6498 df-fv 6504 df-riota 7312 df-ov 7359 df-0g 17322 |
This theorem is referenced by: ress0g 18583 imasmnd2 18592 smndex1id 18720 isgrpde 18770 xrs0 31861 |
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