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Mirrors > Home > MPE Home > Th. List > chrval | Structured version Visualization version GIF version |
Description: Definition substitution of the ring characteristic. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Ref | Expression |
---|---|
chrval.o | ⊢ 𝑂 = (od‘𝑅) |
chrval.u | ⊢ 1 = (1r‘𝑅) |
chrval.c | ⊢ 𝐶 = (chr‘𝑅) |
Ref | Expression |
---|---|
chrval | ⊢ (𝑂‘ 1 ) = 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chrval.c | . 2 ⊢ 𝐶 = (chr‘𝑅) | |
2 | fveq2 6907 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (od‘𝑟) = (od‘𝑅)) | |
3 | chrval.o | . . . . . 6 ⊢ 𝑂 = (od‘𝑅) | |
4 | 2, 3 | eqtr4di 2793 | . . . . 5 ⊢ (𝑟 = 𝑅 → (od‘𝑟) = 𝑂) |
5 | fveq2 6907 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅)) | |
6 | chrval.u | . . . . . 6 ⊢ 1 = (1r‘𝑅) | |
7 | 5, 6 | eqtr4di 2793 | . . . . 5 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 ) |
8 | 4, 7 | fveq12d 6914 | . . . 4 ⊢ (𝑟 = 𝑅 → ((od‘𝑟)‘(1r‘𝑟)) = (𝑂‘ 1 )) |
9 | df-chr 21534 | . . . 4 ⊢ chr = (𝑟 ∈ V ↦ ((od‘𝑟)‘(1r‘𝑟))) | |
10 | fvex 6920 | . . . 4 ⊢ (𝑂‘ 1 ) ∈ V | |
11 | 8, 9, 10 | fvmpt 7016 | . . 3 ⊢ (𝑅 ∈ V → (chr‘𝑅) = (𝑂‘ 1 )) |
12 | fvprc 6899 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (chr‘𝑅) = ∅) | |
13 | fvprc 6899 | . . . . . . 7 ⊢ (¬ 𝑅 ∈ V → (od‘𝑅) = ∅) | |
14 | 3, 13 | eqtrid 2787 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ∅) |
15 | 14 | fveq1d 6909 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝑂‘ 1 ) = (∅‘ 1 )) |
16 | 0fv 6951 | . . . . 5 ⊢ (∅‘ 1 ) = ∅ | |
17 | 15, 16 | eqtrdi 2791 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑂‘ 1 ) = ∅) |
18 | 12, 17 | eqtr4d 2778 | . . 3 ⊢ (¬ 𝑅 ∈ V → (chr‘𝑅) = (𝑂‘ 1 )) |
19 | 11, 18 | pm2.61i 182 | . 2 ⊢ (chr‘𝑅) = (𝑂‘ 1 ) |
20 | 1, 19 | eqtr2i 2764 | 1 ⊢ (𝑂‘ 1 ) = 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 ‘cfv 6563 odcod 19557 1rcur 20199 chrcchr 21530 |
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-sep 5302 ax-nul 5312 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-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 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-iota 6516 df-fun 6565 df-fv 6571 df-chr 21534 |
This theorem is referenced by: chrcl 21557 chrid 21558 chrdvds 21559 chrcong 21560 dvdschrmulg 21561 ply1chr 22326 subrgchr 33227 ofldchr 33324 zrhchr 33937 |
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