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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 18128. (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 7176 | . . . 4 ⊢ (𝜑 → ( 0 + 0 ) = ( 0 (+g‘𝐺) 0 )) |
3 | oveq2 7167 | . . . . . 6 ⊢ (𝑥 = 0 → ( 0 + 𝑥) = ( 0 + 0 )) | |
4 | id 22 | . . . . . 6 ⊢ (𝑥 = 0 → 𝑥 = 0 ) | |
5 | 3, 4 | eqeq12d 2840 | . . . . 5 ⊢ (𝑥 = 0 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 + 0 ) = 0 )) |
6 | grpidd2.i | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) | |
7 | 6 | ralrimiva 3185 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ( 0 + 𝑥) = 𝑥) |
8 | grpidd2.z | . . . . 5 ⊢ (𝜑 → 0 ∈ 𝐵) | |
9 | 5, 7, 8 | rspcdva 3628 | . . . 4 ⊢ (𝜑 → ( 0 + 0 ) = 0 ) |
10 | 2, 9 | eqtr3d 2861 | . . 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 2918 | . . . 4 ⊢ (𝜑 → 0 ∈ (Base‘𝐺)) |
14 | eqid 2824 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
15 | eqid 2824 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
16 | eqid 2824 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
17 | 14, 15, 16 | grpid 18142 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → (( 0 (+g‘𝐺) 0 ) = 0 ↔ (0g‘𝐺) = 0 )) |
18 | 11, 13, 17 | syl2anc 586 | . . 3 ⊢ (𝜑 → (( 0 (+g‘𝐺) 0 ) = 0 ↔ (0g‘𝐺) = 0 )) |
19 | 10, 18 | mpbid 234 | . 2 ⊢ (𝜑 → (0g‘𝐺) = 0 ) |
20 | 19 | eqcomd 2830 | 1 ⊢ (𝜑 → 0 = (0g‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 +gcplusg 16568 0gc0g 16716 Grpcgrp 18106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-riota 7117 df-ov 7162 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 |
This theorem is referenced by: imasgrp2 18217 |
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