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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 4231 | . . . 4 ⊢ ¬ 𝑋 ∈ ∅ | |
2 | cntzrcl.z | . . . . . . . 8 ⊢ 𝑍 = (Cntz‘𝑀) | |
3 | fvprc 6687 | . . . . . . . 8 ⊢ (¬ 𝑀 ∈ V → (Cntz‘𝑀) = ∅) | |
4 | 2, 3 | syl5eq 2783 | . . . . . . 7 ⊢ (¬ 𝑀 ∈ V → 𝑍 = ∅) |
5 | 4 | fveq1d 6697 | . . . . . 6 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝑆) = (∅‘𝑆)) |
6 | 0fv 6734 | . . . . . 6 ⊢ (∅‘𝑆) = ∅ | |
7 | 5, 6 | eqtrdi 2787 | . . . . 5 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝑆) = ∅) |
8 | 7 | eleq2d 2816 | . . . 4 ⊢ (¬ 𝑀 ∈ V → (𝑋 ∈ (𝑍‘𝑆) ↔ 𝑋 ∈ ∅)) |
9 | 1, 8 | mtbiri 330 | . . 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 2736 | . . . . . . . 8 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
13 | 11, 12, 2 | cntzfval 18668 | . . . . . . 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 5759 | . . . . 5 ⊢ (𝑋 ∈ (𝑍‘𝑆) → dom 𝑍 = dom (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝑀)𝑧) = (𝑧(+g‘𝑀)𝑦)})) |
16 | eqid 2736 | . . . . . 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 6084 | . . . . 5 ⊢ dom (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝑀)𝑧) = (𝑧(+g‘𝑀)𝑦)}) ⊆ 𝒫 𝐵 |
18 | 15, 17 | eqsstrdi 3941 | . . . 4 ⊢ (𝑋 ∈ (𝑍‘𝑆) → dom 𝑍 ⊆ 𝒫 𝐵) |
19 | elfvdm 6727 | . . . 4 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑆 ∈ dom 𝑍) | |
20 | 18, 19 | sseldd 3888 | . . 3 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑆 ∈ 𝒫 𝐵) |
21 | 20 | elpwid 4510 | . 2 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑆 ⊆ 𝐵) |
22 | 10, 21 | jca 515 | 1 ⊢ (𝑋 ∈ (𝑍‘𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 {crab 3055 Vcvv 3398 ⊆ wss 3853 ∅c0 4223 𝒫 cpw 4499 ↦ cmpt 5120 dom cdm 5536 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 +gcplusg 16749 Cntzccntz 18663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-cntz 18665 |
This theorem is referenced by: cntzssv 18676 cntzi 18677 resscntz 18680 cntzmhm 18687 oppgcntz 18710 |
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