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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 2735 | . 2 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2735 | . 2 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
3 | eqid 2735 | . 2 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | grpidd.z | . . 3 ⊢ (𝜑 → 0 ∈ 𝐵) | |
5 | grpidd.b | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) | |
6 | 4, 5 | eleqtrd 2841 | . 2 ⊢ (𝜑 → 0 ∈ (Base‘𝐺)) |
7 | 5 | eleq2d 2825 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐺))) |
8 | 7 | biimpar 477 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ 𝐵) |
9 | grpidd.p | . . . . . 6 ⊢ (𝜑 → + = (+g‘𝐺)) | |
10 | 9 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → + = (+g‘𝐺)) |
11 | 10 | oveqd 7448 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = ( 0 (+g‘𝐺)𝑥)) |
12 | grpidd.i | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) | |
13 | 11, 12 | eqtr3d 2777 | . . 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 7448 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = (𝑥(+g‘𝐺) 0 )) |
16 | grpidd.j | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥) | |
17 | 15, 16 | eqtr3d 2777 | . . 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 18694 | 1 ⊢ (𝜑 → 0 = (0g‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 0gc0g 17486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-riota 7388 df-ov 7434 df-0g 17488 |
This theorem is referenced by: ress0g 18788 imasmnd2 18800 smndex1id 18937 isgrpde 18988 xrs0 32991 |
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