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Mirrors > Home > MPE Home > Th. List > elcntzsn | Structured version Visualization version GIF version |
Description: Value of the centralizer of a singleton. (Contributed by Mario Carneiro, 25-Apr-2016.) |
Ref | Expression |
---|---|
cntzfval.b | ⊢ 𝐵 = (Base‘𝑀) |
cntzfval.p | ⊢ + = (+g‘𝑀) |
cntzfval.z | ⊢ 𝑍 = (Cntz‘𝑀) |
Ref | Expression |
---|---|
elcntzsn | ⊢ (𝑌 ∈ 𝐵 → (𝑋 ∈ (𝑍‘{𝑌}) ↔ (𝑋 ∈ 𝐵 ∧ (𝑋 + 𝑌) = (𝑌 + 𝑋)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cntzfval.b | . . . 4 ⊢ 𝐵 = (Base‘𝑀) | |
2 | cntzfval.p | . . . 4 ⊢ + = (+g‘𝑀) | |
3 | cntzfval.z | . . . 4 ⊢ 𝑍 = (Cntz‘𝑀) | |
4 | 1, 2, 3 | cntzsnval 17580 | . . 3 ⊢ (𝑌 ∈ 𝐵 → (𝑍‘{𝑌}) = {𝑥 ∈ 𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)}) |
5 | 4 | eleq2d 2673 | . 2 ⊢ (𝑌 ∈ 𝐵 → (𝑋 ∈ (𝑍‘{𝑌}) ↔ 𝑋 ∈ {𝑥 ∈ 𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)})) |
6 | oveq1 6556 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 + 𝑌) = (𝑋 + 𝑌)) | |
7 | oveq2 6557 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑌 + 𝑥) = (𝑌 + 𝑋)) | |
8 | 6, 7 | eqeq12d 2625 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 + 𝑌) = (𝑌 + 𝑥) ↔ (𝑋 + 𝑌) = (𝑌 + 𝑋))) |
9 | 8 | elrab 3331 | . 2 ⊢ (𝑋 ∈ {𝑥 ∈ 𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)} ↔ (𝑋 ∈ 𝐵 ∧ (𝑋 + 𝑌) = (𝑌 + 𝑋))) |
10 | 5, 9 | syl6bb 275 | 1 ⊢ (𝑌 ∈ 𝐵 → (𝑋 ∈ (𝑍‘{𝑌}) ↔ (𝑋 ∈ 𝐵 ∧ (𝑋 + 𝑌) = (𝑌 + 𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 {csn 4125 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 Cntzccntz 17571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-cntz 17573 |
This theorem is referenced by: gsumconst 18157 gsumpt 18184 |
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