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Mirrors > Home > MPE Home > Th. List > sscntz | Structured version Visualization version GIF version |
Description: A centralizer expression for two sets elementwise commuting. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Ref | Expression |
---|---|
cntzfval.b | ⊢ 𝐵 = (Base‘𝑀) |
cntzfval.p | ⊢ + = (+g‘𝑀) |
cntzfval.z | ⊢ 𝑍 = (Cntz‘𝑀) |
Ref | Expression |
---|---|
sscntz | ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑆 ⊆ (𝑍‘𝑇) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cntzfval.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑀) | |
2 | cntzfval.p | . . . . 5 ⊢ + = (+g‘𝑀) | |
3 | cntzfval.z | . . . . 5 ⊢ 𝑍 = (Cntz‘𝑀) | |
4 | 1, 2, 3 | cntzval 17754 | . . . 4 ⊢ (𝑇 ⊆ 𝐵 → (𝑍‘𝑇) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)}) |
5 | 4 | sseq2d 3633 | . . 3 ⊢ (𝑇 ⊆ 𝐵 → (𝑆 ⊆ (𝑍‘𝑇) ↔ 𝑆 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)})) |
6 | ssrab 3680 | . . 3 ⊢ (𝑆 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ↔ (𝑆 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥))) | |
7 | 5, 6 | syl6bb 276 | . 2 ⊢ (𝑇 ⊆ 𝐵 → (𝑆 ⊆ (𝑍‘𝑇) ↔ (𝑆 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)))) |
8 | ibar 525 | . . 3 ⊢ (𝑆 ⊆ 𝐵 → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑆 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)))) | |
9 | 8 | bicomd 213 | . 2 ⊢ (𝑆 ⊆ 𝐵 → ((𝑆 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥))) |
10 | 7, 9 | sylan9bbr 737 | 1 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑆 ⊆ (𝑍‘𝑇) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∀wral 2912 {crab 2916 ⊆ wss 3574 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 Cntzccntz 17748 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-cntz 17750 |
This theorem is referenced by: cntz2ss 17765 cntzrec 17766 submcmn2 18244 mplcoe5lem 19467 |
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