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Mirrors > Home > MPE Home > Th. List > cntzfval | Structured version Visualization version GIF version |
Description: First level substitution for a centralizer. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Ref | Expression |
---|---|
cntzfval.b | ⊢ 𝐵 = (Base‘𝑀) |
cntzfval.p | ⊢ + = (+g‘𝑀) |
cntzfval.z | ⊢ 𝑍 = (Cntz‘𝑀) |
Ref | Expression |
---|---|
cntzfval | ⊢ (𝑀 ∈ 𝑉 → 𝑍 = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cntzfval.z | . 2 ⊢ 𝑍 = (Cntz‘𝑀) | |
2 | elex 3449 | . . 3 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V) | |
3 | fveq2 6776 | . . . . . . 7 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀)) | |
4 | cntzfval.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑀) | |
5 | 3, 4 | eqtr4di 2796 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵) |
6 | 5 | pweqd 4554 | . . . . 5 ⊢ (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 𝐵) |
7 | fveq2 6776 | . . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀)) | |
8 | cntzfval.p | . . . . . . . . . 10 ⊢ + = (+g‘𝑀) | |
9 | 7, 8 | eqtr4di 2796 | . . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = + ) |
10 | 9 | oveqd 7294 | . . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑥(+g‘𝑚)𝑦) = (𝑥 + 𝑦)) |
11 | 9 | oveqd 7294 | . . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑦(+g‘𝑚)𝑥) = (𝑦 + 𝑥)) |
12 | 10, 11 | eqeq12d 2754 | . . . . . . 7 ⊢ (𝑚 = 𝑀 → ((𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥) ↔ (𝑥 + 𝑦) = (𝑦 + 𝑥))) |
13 | 12 | ralbidv 3119 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥) ↔ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥))) |
14 | 5, 13 | rabeqbidv 3419 | . . . . 5 ⊢ (𝑚 = 𝑀 → {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥)} = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}) |
15 | 6, 14 | mpteq12dv 5167 | . . . 4 ⊢ (𝑚 = 𝑀 → (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥)}) = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})) |
16 | df-cntz 18921 | . . . 4 ⊢ Cntz = (𝑚 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥)})) | |
17 | 4 | fvexi 6790 | . . . . . 6 ⊢ 𝐵 ∈ V |
18 | 17 | pwex 5305 | . . . . 5 ⊢ 𝒫 𝐵 ∈ V |
19 | 18 | mptex 7101 | . . . 4 ⊢ (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}) ∈ V |
20 | 15, 16, 19 | fvmpt 6877 | . . 3 ⊢ (𝑀 ∈ V → (Cntz‘𝑀) = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})) |
21 | 2, 20 | syl 17 | . 2 ⊢ (𝑀 ∈ 𝑉 → (Cntz‘𝑀) = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})) |
22 | 1, 21 | eqtrid 2790 | 1 ⊢ (𝑀 ∈ 𝑉 → 𝑍 = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ∀wral 3064 {crab 3068 Vcvv 3431 𝒫 cpw 4535 ↦ cmpt 5159 ‘cfv 6435 (class class class)co 7277 Basecbs 16910 +gcplusg 16960 Cntzccntz 18919 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5211 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-iun 4928 df-br 5077 df-opab 5139 df-mpt 5160 df-id 5491 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-ov 7280 df-cntz 18921 |
This theorem is referenced by: cntzval 18925 cntzrcl 18931 |
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