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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 3514 | . . 3 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V) | |
3 | fveq2 6672 | . . . . . . 7 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀)) | |
4 | cntzfval.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑀) | |
5 | 3, 4 | syl6eqr 2876 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵) |
6 | 5 | pweqd 4560 | . . . . 5 ⊢ (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 𝐵) |
7 | fveq2 6672 | . . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀)) | |
8 | cntzfval.p | . . . . . . . . . 10 ⊢ + = (+g‘𝑀) | |
9 | 7, 8 | syl6eqr 2876 | . . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = + ) |
10 | 9 | oveqd 7175 | . . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑥(+g‘𝑚)𝑦) = (𝑥 + 𝑦)) |
11 | 9 | oveqd 7175 | . . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑦(+g‘𝑚)𝑥) = (𝑦 + 𝑥)) |
12 | 10, 11 | eqeq12d 2839 | . . . . . . 7 ⊢ (𝑚 = 𝑀 → ((𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥) ↔ (𝑥 + 𝑦) = (𝑦 + 𝑥))) |
13 | 12 | ralbidv 3199 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥) ↔ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥))) |
14 | 5, 13 | rabeqbidv 3487 | . . . . 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 5153 | . . . 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 18449 | . . . 4 ⊢ Cntz = (𝑚 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥)})) | |
17 | 4 | fvexi 6686 | . . . . . 6 ⊢ 𝐵 ∈ V |
18 | 17 | pwex 5283 | . . . . 5 ⊢ 𝒫 𝐵 ∈ V |
19 | 18 | mptex 6988 | . . . 4 ⊢ (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}) ∈ V |
20 | 15, 16, 19 | fvmpt 6770 | . . 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 | syl5eq 2870 | 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 1537 ∈ wcel 2114 ∀wral 3140 {crab 3144 Vcvv 3496 𝒫 cpw 4541 ↦ cmpt 5148 ‘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: cntzval 18453 cntzrcl 18459 |
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