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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 18922. (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 7443 | . . . 4 ⊢ (𝜑 → ( 0 + 0 ) = ( 0 (+g‘𝐺) 0 )) |
3 | oveq2 7434 | . . . . . 6 ⊢ (𝑥 = 0 → ( 0 + 𝑥) = ( 0 + 0 )) | |
4 | id 22 | . . . . . 6 ⊢ (𝑥 = 0 → 𝑥 = 0 ) | |
5 | 3, 4 | eqeq12d 2744 | . . . . 5 ⊢ (𝑥 = 0 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 + 0 ) = 0 )) |
6 | grpidd2.i | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) | |
7 | 6 | ralrimiva 3143 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ( 0 + 𝑥) = 𝑥) |
8 | grpidd2.z | . . . . 5 ⊢ (𝜑 → 0 ∈ 𝐵) | |
9 | 5, 7, 8 | rspcdva 3612 | . . . 4 ⊢ (𝜑 → ( 0 + 0 ) = 0 ) |
10 | 2, 9 | eqtr3d 2770 | . . 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 2831 | . . . 4 ⊢ (𝜑 → 0 ∈ (Base‘𝐺)) |
14 | eqid 2728 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
15 | eqid 2728 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
16 | eqid 2728 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
17 | 14, 15, 16 | grpid 18939 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → (( 0 (+g‘𝐺) 0 ) = 0 ↔ (0g‘𝐺) = 0 )) |
18 | 11, 13, 17 | syl2anc 582 | . . 3 ⊢ (𝜑 → (( 0 (+g‘𝐺) 0 ) = 0 ↔ (0g‘𝐺) = 0 )) |
19 | 10, 18 | mpbid 231 | . 2 ⊢ (𝜑 → (0g‘𝐺) = 0 ) |
20 | 19 | eqcomd 2734 | 1 ⊢ (𝜑 → 0 = (0g‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 +gcplusg 17240 0gc0g 17428 Grpcgrp 18897 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-riota 7382 df-ov 7429 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 |
This theorem is referenced by: imasgrp2 19018 |
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