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| Mirrors > Home > MPE Home > Th. List > grpocl | Structured version Visualization version GIF version | ||
| Description: Closure law for a group operation. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| grpfo.1 | ⊢ 𝑋 = ran 𝐺 |
| Ref | Expression |
|---|---|
| grpocl | ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | grpfo.1 | . . . 4 ⊢ 𝑋 = ran 𝐺 | |
| 2 | 1 | grpofo 30518 | . . 3 ⊢ (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)–onto→𝑋) |
| 3 | fof 6820 | . . 3 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 → 𝐺:(𝑋 × 𝑋)⟶𝑋) | |
| 4 | 2, 3 | syl 17 | . 2 ⊢ (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)⟶𝑋) |
| 5 | fovcdm 7603 | . 2 ⊢ ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋) | |
| 6 | 4, 5 | syl3an1 1164 | 1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 × cxp 5683 ran crn 5686 ⟶wf 6557 –onto→wfo 6559 (class class class)co 7431 GrpOpcgr 30508 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-ov 7434 df-grpo 30512 |
| This theorem is referenced by: grpoidinvlem2 30524 grpoidinvlem3 30525 grpoinvop 30552 grpodivf 30557 grpomuldivass 30560 ablo4 30569 nvgcl 30639 ablo4pnp 37887 ghomco 37898 rngogcl 37919 divrngcl 37964 iscringd 38005 |
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