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Mirrors > Home > MPE Home > Th. List > grpidd2 | Structured version Visualization version GIF version |
Description: Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 18885. (Contributed by Mario Carneiro, 14-Jun-2015.) |
Ref | Expression |
---|---|
grpidd2.b | ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) |
grpidd2.p | ⊢ (𝜑 → + = (+g‘𝐺)) |
grpidd2.z | ⊢ (𝜑 → 0 ∈ 𝐵) |
grpidd2.i | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) |
grpidd2.j | ⊢ (𝜑 → 𝐺 ∈ Grp) |
Ref | Expression |
---|---|
grpidd2 | ⊢ (𝜑 → 0 = (0g‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpidd2.p | . . . . 5 ⊢ (𝜑 → + = (+g‘𝐺)) | |
2 | 1 | oveqd 7421 | . . . 4 ⊢ (𝜑 → ( 0 + 0 ) = ( 0 (+g‘𝐺) 0 )) |
3 | oveq2 7412 | . . . . . 6 ⊢ (𝑥 = 0 → ( 0 + 𝑥) = ( 0 + 0 )) | |
4 | id 22 | . . . . . 6 ⊢ (𝑥 = 0 → 𝑥 = 0 ) | |
5 | 3, 4 | eqeq12d 2742 | . . . . 5 ⊢ (𝑥 = 0 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 + 0 ) = 0 )) |
6 | grpidd2.i | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) | |
7 | 6 | ralrimiva 3140 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ( 0 + 𝑥) = 𝑥) |
8 | grpidd2.z | . . . . 5 ⊢ (𝜑 → 0 ∈ 𝐵) | |
9 | 5, 7, 8 | rspcdva 3607 | . . . 4 ⊢ (𝜑 → ( 0 + 0 ) = 0 ) |
10 | 2, 9 | eqtr3d 2768 | . . 3 ⊢ (𝜑 → ( 0 (+g‘𝐺) 0 ) = 0 ) |
11 | grpidd2.j | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
12 | grpidd2.b | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) | |
13 | 8, 12 | eleqtrd 2829 | . . . 4 ⊢ (𝜑 → 0 ∈ (Base‘𝐺)) |
14 | eqid 2726 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
15 | eqid 2726 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
16 | eqid 2726 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
17 | 14, 15, 16 | grpid 18902 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → (( 0 (+g‘𝐺) 0 ) = 0 ↔ (0g‘𝐺) = 0 )) |
18 | 11, 13, 17 | syl2anc 583 | . . 3 ⊢ (𝜑 → (( 0 (+g‘𝐺) 0 ) = 0 ↔ (0g‘𝐺) = 0 )) |
19 | 10, 18 | mpbid 231 | . 2 ⊢ (𝜑 → (0g‘𝐺) = 0 ) |
20 | 19 | eqcomd 2732 | 1 ⊢ (𝜑 → 0 = (0g‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ‘cfv 6536 (class class class)co 7404 Basecbs 17150 +gcplusg 17203 0gc0g 17391 Grpcgrp 18860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-riota 7360 df-ov 7407 df-0g 17393 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-grp 18863 |
This theorem is referenced by: imasgrp2 18980 |
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