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Mirrors > Home > MPE Home > Th. List > cntzrecd | Structured version Visualization version GIF version |
Description: Commute the "subgroups commute" predicate. (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
cntzrecd.z | ⊢ 𝑍 = (Cntz‘𝐺) |
cntzrecd.t | ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) |
cntzrecd.u | ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) |
cntzrecd.s | ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) |
Ref | Expression |
---|---|
cntzrecd | ⊢ (𝜑 → 𝑈 ⊆ (𝑍‘𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cntzrecd.s | . 2 ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) | |
2 | cntzrecd.t | . . 3 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) | |
3 | cntzrecd.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) | |
4 | eqid 2798 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
5 | 4 | subgss 18272 | . . . 4 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺)) |
6 | 4 | subgss 18272 | . . . 4 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺)) |
7 | cntzrecd.z | . . . . 5 ⊢ 𝑍 = (Cntz‘𝐺) | |
8 | 4, 7 | cntzrec 18456 | . . . 4 ⊢ ((𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 ⊆ (𝑍‘𝑈) ↔ 𝑈 ⊆ (𝑍‘𝑇))) |
9 | 5, 6, 8 | syl2an 598 | . . 3 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊆ (𝑍‘𝑈) ↔ 𝑈 ⊆ (𝑍‘𝑇))) |
10 | 2, 3, 9 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝑇 ⊆ (𝑍‘𝑈) ↔ 𝑈 ⊆ (𝑍‘𝑇))) |
11 | 1, 10 | mpbid 235 | 1 ⊢ (𝜑 → 𝑈 ⊆ (𝑍‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ‘cfv 6324 Basecbs 16475 SubGrpcsubg 18265 Cntzccntz 18437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-subg 18268 df-cntz 18439 |
This theorem is referenced by: subgdisj2 18810 pj2f 18816 pj1id 18817 dprdcntz2 19153 dmdprdsplit2lem 19160 dmdprdsplit2 19161 |
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