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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 2825 | . . . . . . 7 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | cntzi.z | . . . . . . 7 ⊢ 𝑍 = (Cntz‘𝑀) | |
3 | 1, 2 | cntzrcl 18110 | . . . . . 6 ⊢ (𝑋 ∈ (𝑍‘𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ (Base‘𝑀))) |
4 | 3 | simprd 491 | . . . . 5 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑆 ⊆ (Base‘𝑀)) |
5 | cntzi.p | . . . . . 6 ⊢ + = (+g‘𝑀) | |
6 | 1, 5, 2 | elcntz 18105 | . . . . 5 ⊢ (𝑆 ⊆ (Base‘𝑀) → (𝑋 ∈ (𝑍‘𝑆) ↔ (𝑋 ∈ (Base‘𝑀) ∧ ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋)))) |
7 | 4, 6 | syl 17 | . . . 4 ⊢ (𝑋 ∈ (𝑍‘𝑆) → (𝑋 ∈ (𝑍‘𝑆) ↔ (𝑋 ∈ (Base‘𝑀) ∧ ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋)))) |
8 | 7 | simplbda 495 | . . 3 ⊢ ((𝑋 ∈ (𝑍‘𝑆) ∧ 𝑋 ∈ (𝑍‘𝑆)) → ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋)) |
9 | 8 | anidms 564 | . 2 ⊢ (𝑋 ∈ (𝑍‘𝑆) → ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋)) |
10 | oveq2 6913 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌)) | |
11 | oveq1 6912 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 + 𝑋) = (𝑌 + 𝑋)) | |
12 | 10, 11 | eqeq12d 2840 | . . 3 ⊢ (𝑦 = 𝑌 → ((𝑋 + 𝑦) = (𝑦 + 𝑋) ↔ (𝑋 + 𝑌) = (𝑌 + 𝑋))) |
13 | 12 | rspccva 3525 | . 2 ⊢ ((∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋) ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
14 | 9, 13 | sylan 577 | 1 ⊢ ((𝑋 ∈ (𝑍‘𝑆) ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∀wral 3117 Vcvv 3414 ⊆ wss 3798 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 +gcplusg 16305 Cntzccntz 18098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-cntz 18100 |
This theorem is referenced by: cntri 18113 cntz2ss 18115 cntzsubm 18118 cntzsubg 18119 cntzmhm 18121 cntrsubgnsg 18123 lsmsubm 18419 lsmsubg 18420 lsmcom2 18421 subgdisj1 18455 subgdisj2 18456 pj1id 18463 pj1ghm 18467 gsumval3eu 18658 gsumval3 18661 gsumzaddlem 18674 gsumzoppg 18697 dprdfcntz 18768 cntzsubr 19168 cntzsdrg 38615 |
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