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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 2762 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
| 2 | eqid 2762 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
| 3 | gagrpid.1 | . . . . 5 ⊢ 0 = (0g‘𝐺) | |
| 4 | 1, 2, 3 | isga 19331 | . . . 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 501 | . . 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 486 | . . . 4 ⊢ ((( 0 ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))) → ( 0 ⊕ 𝑥) = 𝑥) | |
| 7 | 6 | ralimi 3099 | . . 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 511 | . 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∀𝑥 ∈ 𝑌 ( 0 ⊕ 𝑥) = 𝑥) |
| 9 | oveq2 7404 | . . . 4 ⊢ (𝑥 = 𝐴 → ( 0 ⊕ 𝑥) = ( 0 ⊕ 𝐴)) | |
| 10 | id 22 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
| 11 | 9, 10 | eqeq12d 2778 | . . 3 ⊢ (𝑥 = 𝐴 → (( 0 ⊕ 𝑥) = 𝑥 ↔ ( 0 ⊕ 𝐴) = 𝐴)) |
| 12 | 11 | rspccva 3580 | . 2 ⊢ ((∀𝑥 ∈ 𝑌 ( 0 ⊕ 𝑥) = 𝑥 ∧ 𝐴 ∈ 𝑌) → ( 0 ⊕ 𝐴) = 𝐴) |
| 13 | 8, 12 | sylan 589 | 1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ( 0 ⊕ 𝐴) = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 399 = wceq 1560 ∈ wcel 2142 ∀wral 3076 Vcvv 3454 × cxp 5645 ⟶wf 6517 ‘cfv 6521 (class class class)co 7396 Basecbs 17245 +gcplusg 17286 0gc0g 17468 Grpcgrp 18975 GrpAct cga 19329 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 |
| This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3077 df-rex 3087 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-id 5542 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-fv 6529 df-ov 7399 df-oprab 7400 df-mpo 7401 df-map 8810 df-ga 19330 |
| This theorem is referenced by: gafo 19336 gass 19341 gasubg 19342 galcan 19344 gacan 19345 gaorber 19348 gastacl 19349 |
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