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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 6437 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (od‘𝑟) = (od‘𝑅)) | |
3 | chrval.o | . . . . . 6 ⊢ 𝑂 = (od‘𝑅) | |
4 | 2, 3 | syl6eqr 2879 | . . . . 5 ⊢ (𝑟 = 𝑅 → (od‘𝑟) = 𝑂) |
5 | fveq2 6437 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅)) | |
6 | chrval.u | . . . . . 6 ⊢ 1 = (1r‘𝑅) | |
7 | 5, 6 | syl6eqr 2879 | . . . . 5 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 ) |
8 | 4, 7 | fveq12d 6444 | . . . 4 ⊢ (𝑟 = 𝑅 → ((od‘𝑟)‘(1r‘𝑟)) = (𝑂‘ 1 )) |
9 | df-chr 20221 | . . . 4 ⊢ chr = (𝑟 ∈ V ↦ ((od‘𝑟)‘(1r‘𝑟))) | |
10 | fvex 6450 | . . . 4 ⊢ (𝑂‘ 1 ) ∈ V | |
11 | 8, 9, 10 | fvmpt 6533 | . . 3 ⊢ (𝑅 ∈ V → (chr‘𝑅) = (𝑂‘ 1 )) |
12 | fvprc 6430 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (chr‘𝑅) = ∅) | |
13 | fvprc 6430 | . . . . . . 7 ⊢ (¬ 𝑅 ∈ V → (od‘𝑅) = ∅) | |
14 | 3, 13 | syl5eq 2873 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ∅) |
15 | 14 | fveq1d 6439 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝑂‘ 1 ) = (∅‘ 1 )) |
16 | 0fv 6477 | . . . . 5 ⊢ (∅‘ 1 ) = ∅ | |
17 | 15, 16 | syl6eq 2877 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑂‘ 1 ) = ∅) |
18 | 12, 17 | eqtr4d 2864 | . . 3 ⊢ (¬ 𝑅 ∈ V → (chr‘𝑅) = (𝑂‘ 1 )) |
19 | 11, 18 | pm2.61i 177 | . 2 ⊢ (chr‘𝑅) = (𝑂‘ 1 ) |
20 | 1, 19 | eqtr2i 2850 | 1 ⊢ (𝑂‘ 1 ) = 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ∅c0 4146 ‘cfv 6127 odcod 18302 1rcur 18862 chrcchr 20217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-iota 6090 df-fun 6129 df-fv 6135 df-chr 20221 |
This theorem is referenced by: chrcl 20241 chrid 20242 chrdvds 20243 chrcong 20244 subrgchr 30335 ofldchr 30355 zrhchr 30561 |
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