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Mirrors > Home > MPE Home > Th. List > cntzrcl | Structured version Visualization version GIF version |
Description: Reverse closure for elements of the centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
Ref | Expression |
---|---|
cntzrcl.b | ⊢ 𝐵 = (Base‘𝑀) |
cntzrcl.z | ⊢ 𝑍 = (Cntz‘𝑀) |
Ref | Expression |
---|---|
cntzrcl | ⊢ (𝑋 ∈ (𝑍‘𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3878 | . . . 4 ⊢ ¬ 𝑋 ∈ ∅ | |
2 | cntzrcl.z | . . . . . . . 8 ⊢ 𝑍 = (Cntz‘𝑀) | |
3 | fvprc 6097 | . . . . . . . 8 ⊢ (¬ 𝑀 ∈ V → (Cntz‘𝑀) = ∅) | |
4 | 2, 3 | syl5eq 2656 | . . . . . . 7 ⊢ (¬ 𝑀 ∈ V → 𝑍 = ∅) |
5 | 4 | fveq1d 6105 | . . . . . 6 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝑆) = (∅‘𝑆)) |
6 | 0fv 6137 | . . . . . 6 ⊢ (∅‘𝑆) = ∅ | |
7 | 5, 6 | syl6eq 2660 | . . . . 5 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝑆) = ∅) |
8 | 7 | eleq2d 2673 | . . . 4 ⊢ (¬ 𝑀 ∈ V → (𝑋 ∈ (𝑍‘𝑆) ↔ 𝑋 ∈ ∅)) |
9 | 1, 8 | mtbiri 316 | . . 3 ⊢ (¬ 𝑀 ∈ V → ¬ 𝑋 ∈ (𝑍‘𝑆)) |
10 | 9 | con4i 112 | . 2 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑀 ∈ V) |
11 | cntzrcl.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑀) | |
12 | eqid 2610 | . . . . . . . 8 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
13 | 11, 12, 2 | cntzfval 17576 | . . . . . . 7 ⊢ (𝑀 ∈ V → 𝑍 = (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝑀)𝑧) = (𝑧(+g‘𝑀)𝑦)})) |
14 | 10, 13 | syl 17 | . . . . . 6 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑍 = (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝑀)𝑧) = (𝑧(+g‘𝑀)𝑦)})) |
15 | 14 | dmeqd 5248 | . . . . 5 ⊢ (𝑋 ∈ (𝑍‘𝑆) → dom 𝑍 = dom (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝑀)𝑧) = (𝑧(+g‘𝑀)𝑦)})) |
16 | eqid 2610 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝑀)𝑧) = (𝑧(+g‘𝑀)𝑦)}) = (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝑀)𝑧) = (𝑧(+g‘𝑀)𝑦)}) | |
17 | 16 | dmmptss 5548 | . . . . 5 ⊢ dom (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝑀)𝑧) = (𝑧(+g‘𝑀)𝑦)}) ⊆ 𝒫 𝐵 |
18 | 15, 17 | syl6eqss 3618 | . . . 4 ⊢ (𝑋 ∈ (𝑍‘𝑆) → dom 𝑍 ⊆ 𝒫 𝐵) |
19 | elfvdm 6130 | . . . 4 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑆 ∈ dom 𝑍) | |
20 | 18, 19 | sseldd 3569 | . . 3 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑆 ∈ 𝒫 𝐵) |
21 | 20 | elpwid 4118 | . 2 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑆 ⊆ 𝐵) |
22 | 10, 21 | jca 553 | 1 ⊢ (𝑋 ∈ (𝑍‘𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 ↦ cmpt 4643 dom cdm 5038 ‘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: cntzssv 17584 cntzi 17585 resscntz 17587 cntzmhm 17594 oppgcntz 17617 |
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