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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 6663 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (od‘𝑟) = (od‘𝑅)) | |
3 | chrval.o | . . . . . 6 ⊢ 𝑂 = (od‘𝑅) | |
4 | 2, 3 | syl6eqr 2871 | . . . . 5 ⊢ (𝑟 = 𝑅 → (od‘𝑟) = 𝑂) |
5 | fveq2 6663 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅)) | |
6 | chrval.u | . . . . . 6 ⊢ 1 = (1r‘𝑅) | |
7 | 5, 6 | syl6eqr 2871 | . . . . 5 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 ) |
8 | 4, 7 | fveq12d 6670 | . . . 4 ⊢ (𝑟 = 𝑅 → ((od‘𝑟)‘(1r‘𝑟)) = (𝑂‘ 1 )) |
9 | df-chr 20581 | . . . 4 ⊢ chr = (𝑟 ∈ V ↦ ((od‘𝑟)‘(1r‘𝑟))) | |
10 | fvex 6676 | . . . 4 ⊢ (𝑂‘ 1 ) ∈ V | |
11 | 8, 9, 10 | fvmpt 6761 | . . 3 ⊢ (𝑅 ∈ V → (chr‘𝑅) = (𝑂‘ 1 )) |
12 | fvprc 6656 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (chr‘𝑅) = ∅) | |
13 | fvprc 6656 | . . . . . . 7 ⊢ (¬ 𝑅 ∈ V → (od‘𝑅) = ∅) | |
14 | 3, 13 | syl5eq 2865 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ∅) |
15 | 14 | fveq1d 6665 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝑂‘ 1 ) = (∅‘ 1 )) |
16 | 0fv 6702 | . . . . 5 ⊢ (∅‘ 1 ) = ∅ | |
17 | 15, 16 | syl6eq 2869 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑂‘ 1 ) = ∅) |
18 | 12, 17 | eqtr4d 2856 | . . 3 ⊢ (¬ 𝑅 ∈ V → (chr‘𝑅) = (𝑂‘ 1 )) |
19 | 11, 18 | pm2.61i 183 | . 2 ⊢ (chr‘𝑅) = (𝑂‘ 1 ) |
20 | 1, 19 | eqtr2i 2842 | 1 ⊢ (𝑂‘ 1 ) = 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∅c0 4288 ‘cfv 6348 odcod 18581 1rcur 19180 chrcchr 20577 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-chr 20581 |
This theorem is referenced by: chrcl 20601 chrid 20602 chrdvds 20603 chrcong 20604 dvdschrmulg 30785 subrgchr 30792 ofldchr 30814 zrhchr 31116 |
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