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Mirrors > Home > ILE Home > Th. List > grpidd2 | GIF version |
Description: Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 12776. (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 5885 | . . . 4 ⊢ (𝜑 → ( 0 + 0 ) = ( 0 (+g‘𝐺) 0 )) |
3 | oveq2 5876 | . . . . . 6 ⊢ (𝑥 = 0 → ( 0 + 𝑥) = ( 0 + 0 )) | |
4 | id 19 | . . . . . 6 ⊢ (𝑥 = 0 → 𝑥 = 0 ) | |
5 | 3, 4 | eqeq12d 2192 | . . . . 5 ⊢ (𝑥 = 0 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 + 0 ) = 0 )) |
6 | grpidd2.i | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) | |
7 | 6 | ralrimiva 2550 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ( 0 + 𝑥) = 𝑥) |
8 | grpidd2.z | . . . . 5 ⊢ (𝜑 → 0 ∈ 𝐵) | |
9 | 5, 7, 8 | rspcdva 2846 | . . . 4 ⊢ (𝜑 → ( 0 + 0 ) = 0 ) |
10 | 2, 9 | eqtr3d 2212 | . . 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 2256 | . . . 4 ⊢ (𝜑 → 0 ∈ (Base‘𝐺)) |
14 | eqid 2177 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
15 | eqid 2177 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
16 | eqid 2177 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
17 | 14, 15, 16 | grpid 12789 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ (Base‘𝐺)) → (( 0 (+g‘𝐺) 0 ) = 0 ↔ (0g‘𝐺) = 0 )) |
18 | 11, 13, 17 | syl2anc 411 | . . 3 ⊢ (𝜑 → (( 0 (+g‘𝐺) 0 ) = 0 ↔ (0g‘𝐺) = 0 )) |
19 | 10, 18 | mpbid 147 | . 2 ⊢ (𝜑 → (0g‘𝐺) = 0 ) |
20 | 19 | eqcomd 2183 | 1 ⊢ (𝜑 → 0 = (0g‘𝐺)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ‘cfv 5211 (class class class)co 5868 Basecbs 12432 +gcplusg 12505 0gc0g 12640 Grpcgrp 12754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-cnex 7880 ax-resscn 7881 ax-1re 7883 ax-addrcl 7886 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-iota 5173 df-fun 5213 df-fn 5214 df-fv 5219 df-riota 5824 df-ov 5871 df-inn 8896 df-2 8954 df-ndx 12435 df-slot 12436 df-base 12438 df-plusg 12518 df-0g 12642 df-mgm 12654 df-sgrp 12687 df-mnd 12697 df-grp 12757 |
This theorem is referenced by: (None) |
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