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| Mirrors > Home > MPE Home > Th. List > cntzi | Structured version Visualization version GIF version | ||
| Description: Membership in a centralizer (inference). (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 22-Sep-2015.) |
| Ref | Expression |
|---|---|
| cntzi.p | ⊢ + = (+g‘𝑀) |
| cntzi.z | ⊢ 𝑍 = (Cntz‘𝑀) |
| Ref | Expression |
|---|---|
| cntzi | ⊢ ((𝑋 ∈ (𝑍‘𝑆) ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
| 2 | cntzi.z | . . . . . 6 ⊢ 𝑍 = (Cntz‘𝑀) | |
| 3 | 1, 2 | cntzrcl 19345 | . . . . 5 ⊢ (𝑋 ∈ (𝑍‘𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ (Base‘𝑀))) |
| 4 | cntzi.p | . . . . . 6 ⊢ + = (+g‘𝑀) | |
| 5 | 1, 4, 2 | elcntz 19340 | . . . . 5 ⊢ (𝑆 ⊆ (Base‘𝑀) → (𝑋 ∈ (𝑍‘𝑆) ↔ (𝑋 ∈ (Base‘𝑀) ∧ ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋)))) |
| 6 | 3, 5 | simpl2im 503 | . . . 4 ⊢ (𝑋 ∈ (𝑍‘𝑆) → (𝑋 ∈ (𝑍‘𝑆) ↔ (𝑋 ∈ (Base‘𝑀) ∧ ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋)))) |
| 7 | 6 | simplbda 499 | . . 3 ⊢ ((𝑋 ∈ (𝑍‘𝑆) ∧ 𝑋 ∈ (𝑍‘𝑆)) → ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋)) |
| 8 | 7 | anidms 566 | . 2 ⊢ (𝑋 ∈ (𝑍‘𝑆) → ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋)) |
| 9 | oveq2 7439 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌)) | |
| 10 | oveq1 7438 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 + 𝑋) = (𝑌 + 𝑋)) | |
| 11 | 9, 10 | eqeq12d 2753 | . . 3 ⊢ (𝑦 = 𝑌 → ((𝑋 + 𝑦) = (𝑦 + 𝑋) ↔ (𝑋 + 𝑌) = (𝑌 + 𝑋))) |
| 12 | 11 | rspccva 3621 | . 2 ⊢ ((∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋) ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
| 13 | 8, 12 | sylan 580 | 1 ⊢ ((𝑋 ∈ (𝑍‘𝑆) ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ⊆ wss 3951 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Cntzccntz 19333 |
| 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| 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-reu 3381 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-pw 4602 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-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-cntz 19335 |
| This theorem is referenced by: cntri 19350 cntzsgrpcl 19352 cntz2ss 19353 cntzsubm 19356 cntzsubg 19357 cntzmhm 19359 cntrsubgnsg 19361 lsmsubm 19671 lsmsubg 19672 lsmcom2 19673 subgdisj1 19709 subgdisj2 19710 pj1id 19717 pj1ghm 19721 gsumval3eu 19922 gsumval3 19925 gsumzaddlem 19939 gsumzoppg 19962 dprdfcntz 20035 cntzsubrng 20567 cntzsubr 20606 cntzsdrg 20803 |
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