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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 18721 | . . . . 5 ⊢ (𝑋 ∈ (𝑍‘𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ (Base‘𝑀))) |
4 | cntzi.p | . . . . . 6 ⊢ + = (+g‘𝑀) | |
5 | 1, 4, 2 | elcntz 18716 | . . . . 5 ⊢ (𝑆 ⊆ (Base‘𝑀) → (𝑋 ∈ (𝑍‘𝑆) ↔ (𝑋 ∈ (Base‘𝑀) ∧ ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋)))) |
6 | 3, 5 | simpl2im 507 | . . . 4 ⊢ (𝑋 ∈ (𝑍‘𝑆) → (𝑋 ∈ (𝑍‘𝑆) ↔ (𝑋 ∈ (Base‘𝑀) ∧ ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋)))) |
7 | 6 | simplbda 503 | . . 3 ⊢ ((𝑋 ∈ (𝑍‘𝑆) ∧ 𝑋 ∈ (𝑍‘𝑆)) → ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋)) |
8 | 7 | anidms 570 | . 2 ⊢ (𝑋 ∈ (𝑍‘𝑆) → ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋)) |
9 | oveq2 7221 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌)) | |
10 | oveq1 7220 | . . . 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 3536 | . 2 ⊢ ((∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋) ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
13 | 8, 12 | sylan 583 | 1 ⊢ ((𝑋 ∈ (𝑍‘𝑆) ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 Vcvv 3408 ⊆ wss 3866 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 +gcplusg 16802 Cntzccntz 18709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-cntz 18711 |
This theorem is referenced by: cntri 18725 cntz2ss 18727 cntzsubm 18730 cntzsubg 18731 cntzmhm 18733 cntrsubgnsg 18735 lsmsubm 19042 lsmsubg 19043 lsmcom2 19044 subgdisj1 19081 subgdisj2 19082 pj1id 19089 pj1ghm 19093 gsumval3eu 19289 gsumval3 19292 gsumzaddlem 19306 gsumzoppg 19329 dprdfcntz 19402 cntzsubr 19833 cntzsdrg 19846 |
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