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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 4298 | . . . 4 ⊢ ¬ 𝑋 ∈ ∅ | |
2 | cntzrcl.z | . . . . . . . 8 ⊢ 𝑍 = (Cntz‘𝑀) | |
3 | fvprc 6665 | . . . . . . . 8 ⊢ (¬ 𝑀 ∈ V → (Cntz‘𝑀) = ∅) | |
4 | 2, 3 | syl5eq 2870 | . . . . . . 7 ⊢ (¬ 𝑀 ∈ V → 𝑍 = ∅) |
5 | 4 | fveq1d 6674 | . . . . . 6 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝑆) = (∅‘𝑆)) |
6 | 0fv 6711 | . . . . . 6 ⊢ (∅‘𝑆) = ∅ | |
7 | 5, 6 | syl6eq 2874 | . . . . 5 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝑆) = ∅) |
8 | 7 | eleq2d 2900 | . . . 4 ⊢ (¬ 𝑀 ∈ V → (𝑋 ∈ (𝑍‘𝑆) ↔ 𝑋 ∈ ∅)) |
9 | 1, 8 | mtbiri 329 | . . 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 2823 | . . . . . . . 8 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
13 | 11, 12, 2 | cntzfval 18452 | . . . . . . 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 5776 | . . . . 5 ⊢ (𝑋 ∈ (𝑍‘𝑆) → dom 𝑍 = dom (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝑀)𝑧) = (𝑧(+g‘𝑀)𝑦)})) |
16 | eqid 2823 | . . . . . 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 6097 | . . . . 5 ⊢ dom (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝑀)𝑧) = (𝑧(+g‘𝑀)𝑦)}) ⊆ 𝒫 𝐵 |
18 | 15, 17 | eqsstrdi 4023 | . . . 4 ⊢ (𝑋 ∈ (𝑍‘𝑆) → dom 𝑍 ⊆ 𝒫 𝐵) |
19 | elfvdm 6704 | . . . 4 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑆 ∈ dom 𝑍) | |
20 | 18, 19 | sseldd 3970 | . . 3 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑆 ∈ 𝒫 𝐵) |
21 | 20 | elpwid 4552 | . 2 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑆 ⊆ 𝐵) |
22 | 10, 21 | jca 514 | 1 ⊢ (𝑋 ∈ (𝑍‘𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 {crab 3144 Vcvv 3496 ⊆ wss 3938 ∅c0 4293 𝒫 cpw 4541 ↦ cmpt 5148 dom cdm 5557 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 Cntzccntz 18447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-cntz 18449 |
This theorem is referenced by: cntzssv 18460 cntzi 18461 resscntz 18464 cntzmhm 18471 oppgcntz 18494 |
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