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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 4344 | . . . 4 ⊢ ¬ 𝑋 ∈ ∅ | |
2 | cntzrcl.z | . . . . . . . 8 ⊢ 𝑍 = (Cntz‘𝑀) | |
3 | fvprc 6899 | . . . . . . . 8 ⊢ (¬ 𝑀 ∈ V → (Cntz‘𝑀) = ∅) | |
4 | 2, 3 | eqtrid 2787 | . . . . . . 7 ⊢ (¬ 𝑀 ∈ V → 𝑍 = ∅) |
5 | 4 | fveq1d 6909 | . . . . . 6 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝑆) = (∅‘𝑆)) |
6 | 0fv 6951 | . . . . . 6 ⊢ (∅‘𝑆) = ∅ | |
7 | 5, 6 | eqtrdi 2791 | . . . . 5 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝑆) = ∅) |
8 | 7 | eleq2d 2825 | . . . 4 ⊢ (¬ 𝑀 ∈ V → (𝑋 ∈ (𝑍‘𝑆) ↔ 𝑋 ∈ ∅)) |
9 | 1, 8 | mtbiri 327 | . . 3 ⊢ (¬ 𝑀 ∈ V → ¬ 𝑋 ∈ (𝑍‘𝑆)) |
10 | 9 | con4i 114 | . 2 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑀 ∈ V) |
11 | cntzrcl.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑀) | |
12 | eqid 2735 | . . . . . . . 8 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
13 | 11, 12, 2 | cntzfval 19351 | . . . . . . 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 5919 | . . . . 5 ⊢ (𝑋 ∈ (𝑍‘𝑆) → dom 𝑍 = dom (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝑀)𝑧) = (𝑧(+g‘𝑀)𝑦)})) |
16 | eqid 2735 | . . . . . 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 6263 | . . . . 5 ⊢ dom (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝑀)𝑧) = (𝑧(+g‘𝑀)𝑦)}) ⊆ 𝒫 𝐵 |
18 | 15, 17 | eqsstrdi 4050 | . . . 4 ⊢ (𝑋 ∈ (𝑍‘𝑆) → dom 𝑍 ⊆ 𝒫 𝐵) |
19 | elfvdm 6944 | . . . 4 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑆 ∈ dom 𝑍) | |
20 | 18, 19 | sseldd 3996 | . . 3 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑆 ∈ 𝒫 𝐵) |
21 | 20 | elpwid 4614 | . 2 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑆 ⊆ 𝐵) |
22 | 10, 21 | jca 511 | 1 ⊢ (𝑋 ∈ (𝑍‘𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 Vcvv 3478 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 ↦ cmpt 5231 dom cdm 5689 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 Cntzccntz 19346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-cntz 19348 |
This theorem is referenced by: cntzssv 19359 cntzi 19360 resscntz 19364 cntzmhm 19372 oppgcntz 19398 |
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