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Mirrors > Home > MPE Home > Th. List > gaf | Structured version Visualization version GIF version |
Description: The mapping of the group action operation. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.) |
Ref | Expression |
---|---|
gaf.1 | ⊢ 𝑋 = (Base‘𝐺) |
Ref | Expression |
---|---|
gaf | ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gaf.1 | . . . 4 ⊢ 𝑋 = (Base‘𝐺) | |
2 | eqid 2738 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | eqid 2738 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
4 | 1, 2, 3 | isga 18539 | . . 3 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( ⊕ :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 (((0g‘𝐺) ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦(+g‘𝐺)𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))))) |
5 | 4 | simprbi 500 | . 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ( ⊕ :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 (((0g‘𝐺) ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦(+g‘𝐺)𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))))) |
6 | 5 | simpld 498 | 1 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∀wral 3053 Vcvv 3398 × cxp 5523 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 Basecbs 16586 +gcplusg 16668 0gc0g 16816 Grpcgrp 18219 GrpAct cga 18537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-fv 6347 df-ov 7173 df-oprab 7174 df-mpo 7175 df-map 8439 df-ga 18538 |
This theorem is referenced by: gafo 18544 gass 18549 gasubg 18550 gacan 18553 gapm 18554 gastacos 18558 orbsta 18561 galactghm 18650 sylow2alem2 18861 |
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