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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 2818 | . . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | cntzi.z | . . . . . 6 ⊢ 𝑍 = (Cntz‘𝑀) | |
3 | 1, 2 | cntzrcl 18395 | . . . . 5 ⊢ (𝑋 ∈ (𝑍‘𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ (Base‘𝑀))) |
4 | cntzi.p | . . . . . 6 ⊢ + = (+g‘𝑀) | |
5 | 1, 4, 2 | elcntz 18390 | . . . . 5 ⊢ (𝑆 ⊆ (Base‘𝑀) → (𝑋 ∈ (𝑍‘𝑆) ↔ (𝑋 ∈ (Base‘𝑀) ∧ ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋)))) |
6 | 3, 5 | simpl2im 504 | . . . 4 ⊢ (𝑋 ∈ (𝑍‘𝑆) → (𝑋 ∈ (𝑍‘𝑆) ↔ (𝑋 ∈ (Base‘𝑀) ∧ ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋)))) |
7 | 6 | simplbda 500 | . . 3 ⊢ ((𝑋 ∈ (𝑍‘𝑆) ∧ 𝑋 ∈ (𝑍‘𝑆)) → ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋)) |
8 | 7 | anidms 567 | . 2 ⊢ (𝑋 ∈ (𝑍‘𝑆) → ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋)) |
9 | oveq2 7153 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌)) | |
10 | oveq1 7152 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 + 𝑋) = (𝑌 + 𝑋)) | |
11 | 9, 10 | eqeq12d 2834 | . . 3 ⊢ (𝑦 = 𝑌 → ((𝑋 + 𝑦) = (𝑦 + 𝑋) ↔ (𝑋 + 𝑌) = (𝑌 + 𝑋))) |
12 | 11 | rspccva 3619 | . 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 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 ⊆ wss 3933 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 Cntzccntz 18383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-cntz 18385 |
This theorem is referenced by: cntri 18399 cntz2ss 18401 cntzsubm 18404 cntzsubg 18405 cntzmhm 18407 cntrsubgnsg 18409 lsmsubm 18707 lsmsubg 18708 lsmcom2 18709 subgdisj1 18746 subgdisj2 18747 pj1id 18754 pj1ghm 18758 gsumval3eu 18953 gsumval3 18956 gsumzaddlem 18970 gsumzoppg 18993 dprdfcntz 19066 cntzsubr 19497 cntzsdrg 19510 |
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