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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 18989. (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 7448 | . . . 4 ⊢ (𝜑 → ( 0 + 0 ) = ( 0 (+g‘𝐺) 0 )) |
3 | oveq2 7439 | . . . . . 6 ⊢ (𝑥 = 0 → ( 0 + 𝑥) = ( 0 + 0 )) | |
4 | id 22 | . . . . . 6 ⊢ (𝑥 = 0 → 𝑥 = 0 ) | |
5 | 3, 4 | eqeq12d 2751 | . . . . 5 ⊢ (𝑥 = 0 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 + 0 ) = 0 )) |
6 | grpidd2.i | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) | |
7 | 6 | ralrimiva 3144 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ( 0 + 𝑥) = 𝑥) |
8 | grpidd2.z | . . . . 5 ⊢ (𝜑 → 0 ∈ 𝐵) | |
9 | 5, 7, 8 | rspcdva 3623 | . . . 4 ⊢ (𝜑 → ( 0 + 0 ) = 0 ) |
10 | 2, 9 | eqtr3d 2777 | . . 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 2841 | . . . 4 ⊢ (𝜑 → 0 ∈ (Base‘𝐺)) |
14 | eqid 2735 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
15 | eqid 2735 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
16 | eqid 2735 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
17 | 14, 15, 16 | grpid 19006 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → (( 0 (+g‘𝐺) 0 ) = 0 ↔ (0g‘𝐺) = 0 )) |
18 | 11, 13, 17 | syl2anc 584 | . . 3 ⊢ (𝜑 → (( 0 (+g‘𝐺) 0 ) = 0 ↔ (0g‘𝐺) = 0 )) |
19 | 10, 18 | mpbid 232 | . 2 ⊢ (𝜑 → (0g‘𝐺) = 0 ) |
20 | 19 | eqcomd 2741 | 1 ⊢ (𝜑 → 0 = (0g‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 0gc0g 17486 Grpcgrp 18964 |
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 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 |
This theorem is referenced by: imasgrp2 19086 |
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