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Mirrors > Home > MPE Home > Th. List > gagrpid | Structured version Visualization version GIF version |
Description: The identity of the group does not alter the base set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.) |
Ref | Expression |
---|---|
gagrpid.1 | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
gagrpid | ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ( 0 ⊕ 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2820 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2820 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | gagrpid.1 | . . . . 5 ⊢ 0 = (0g‘𝐺) | |
4 | 1, 2, 3 | isga 18416 | . . . 4 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( ⊕ :((Base‘𝐺) × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 (( 0 ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))))) |
5 | 4 | simprbi 499 | . . 3 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ( ⊕ :((Base‘𝐺) × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 (( 0 ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))))) |
6 | simpl 485 | . . . 4 ⊢ ((( 0 ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))) → ( 0 ⊕ 𝑥) = 𝑥) | |
7 | 6 | ralimi 3159 | . . 3 ⊢ (∀𝑥 ∈ 𝑌 (( 0 ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))) → ∀𝑥 ∈ 𝑌 ( 0 ⊕ 𝑥) = 𝑥) |
8 | 5, 7 | simpl2im 506 | . 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∀𝑥 ∈ 𝑌 ( 0 ⊕ 𝑥) = 𝑥) |
9 | oveq2 7157 | . . . 4 ⊢ (𝑥 = 𝐴 → ( 0 ⊕ 𝑥) = ( 0 ⊕ 𝐴)) | |
10 | id 22 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
11 | 9, 10 | eqeq12d 2836 | . . 3 ⊢ (𝑥 = 𝐴 → (( 0 ⊕ 𝑥) = 𝑥 ↔ ( 0 ⊕ 𝐴) = 𝐴)) |
12 | 11 | rspccva 3619 | . 2 ⊢ ((∀𝑥 ∈ 𝑌 ( 0 ⊕ 𝑥) = 𝑥 ∧ 𝐴 ∈ 𝑌) → ( 0 ⊕ 𝐴) = 𝐴) |
13 | 8, 12 | sylan 582 | 1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ( 0 ⊕ 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3137 Vcvv 3491 × cxp 5546 ⟶wf 6344 ‘cfv 6348 (class class class)co 7149 Basecbs 16478 +gcplusg 16560 0gc0g 16708 Grpcgrp 18098 GrpAct cga 18414 |
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 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7152 df-oprab 7153 df-mpo 7154 df-map 8401 df-ga 18415 |
This theorem is referenced by: gafo 18421 gass 18426 gasubg 18427 galcan 18429 gacan 18430 gaorber 18433 gastacl 18434 |
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