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Mirrors > Home > MPE Home > Th. List > cntzrec | Structured version Visualization version GIF version |
Description: Reciprocity relationship for centralizers. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Ref | Expression |
---|---|
cntzrec.b | ⊢ 𝐵 = (Base‘𝑀) |
cntzrec.z | ⊢ 𝑍 = (Cntz‘𝑀) |
Ref | Expression |
---|---|
cntzrec | ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑆 ⊆ (𝑍‘𝑇) ↔ 𝑇 ⊆ (𝑍‘𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralcom 3315 | . . . 4 ⊢ (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥) ↔ ∀𝑦 ∈ 𝑇 ∀𝑥 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)) | |
2 | eqcom 2802 | . . . . 5 ⊢ ((𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥) ↔ (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)) | |
3 | 2 | 2ralbii 3133 | . . . 4 ⊢ (∀𝑦 ∈ 𝑇 ∀𝑥 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥) ↔ ∀𝑦 ∈ 𝑇 ∀𝑥 ∈ 𝑆 (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)) |
4 | 1, 3 | bitri 276 | . . 3 ⊢ (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥) ↔ ∀𝑦 ∈ 𝑇 ∀𝑥 ∈ 𝑆 (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)) |
5 | 4 | a1i 11 | . 2 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥) ↔ ∀𝑦 ∈ 𝑇 ∀𝑥 ∈ 𝑆 (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))) |
6 | cntzrec.b | . . 3 ⊢ 𝐵 = (Base‘𝑀) | |
7 | eqid 2795 | . . 3 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
8 | cntzrec.z | . . 3 ⊢ 𝑍 = (Cntz‘𝑀) | |
9 | 6, 7, 8 | sscntz 18197 | . 2 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑆 ⊆ (𝑍‘𝑇) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥))) |
10 | 6, 7, 8 | sscntz 18197 | . . 3 ⊢ ((𝑇 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝐵) → (𝑇 ⊆ (𝑍‘𝑆) ↔ ∀𝑦 ∈ 𝑇 ∀𝑥 ∈ 𝑆 (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))) |
11 | 10 | ancoms 459 | . 2 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑇 ⊆ (𝑍‘𝑆) ↔ ∀𝑦 ∈ 𝑇 ∀𝑥 ∈ 𝑆 (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦))) |
12 | 5, 9, 11 | 3bitr4d 312 | 1 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑆 ⊆ (𝑍‘𝑇) ↔ 𝑇 ⊆ (𝑍‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ∀wral 3105 ⊆ wss 3859 ‘cfv 6225 (class class class)co 7016 Basecbs 16312 +gcplusg 16394 Cntzccntz 18186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-ov 7019 df-cntz 18188 |
This theorem is referenced by: cntzrecd 18531 lsmcntzr 18533 cntzspan 18687 dprdfadd 18859 |
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