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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 6191 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (od‘𝑟) = (od‘𝑅)) | |
3 | chrval.o | . . . . . 6 ⊢ 𝑂 = (od‘𝑅) | |
4 | 2, 3 | syl6eqr 2674 | . . . . 5 ⊢ (𝑟 = 𝑅 → (od‘𝑟) = 𝑂) |
5 | fveq2 6191 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅)) | |
6 | chrval.u | . . . . . 6 ⊢ 1 = (1r‘𝑅) | |
7 | 5, 6 | syl6eqr 2674 | . . . . 5 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 ) |
8 | 4, 7 | fveq12d 6197 | . . . 4 ⊢ (𝑟 = 𝑅 → ((od‘𝑟)‘(1r‘𝑟)) = (𝑂‘ 1 )) |
9 | df-chr 19854 | . . . 4 ⊢ chr = (𝑟 ∈ V ↦ ((od‘𝑟)‘(1r‘𝑟))) | |
10 | fvex 6201 | . . . 4 ⊢ (𝑂‘ 1 ) ∈ V | |
11 | 8, 9, 10 | fvmpt 6282 | . . 3 ⊢ (𝑅 ∈ V → (chr‘𝑅) = (𝑂‘ 1 )) |
12 | fvprc 6185 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (chr‘𝑅) = ∅) | |
13 | fvprc 6185 | . . . . . . 7 ⊢ (¬ 𝑅 ∈ V → (od‘𝑅) = ∅) | |
14 | 3, 13 | syl5eq 2668 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ∅) |
15 | 14 | fveq1d 6193 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝑂‘ 1 ) = (∅‘ 1 )) |
16 | 0fv 6227 | . . . . 5 ⊢ (∅‘ 1 ) = ∅ | |
17 | 15, 16 | syl6eq 2672 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑂‘ 1 ) = ∅) |
18 | 12, 17 | eqtr4d 2659 | . . 3 ⊢ (¬ 𝑅 ∈ V → (chr‘𝑅) = (𝑂‘ 1 )) |
19 | 11, 18 | pm2.61i 176 | . 2 ⊢ (chr‘𝑅) = (𝑂‘ 1 ) |
20 | 1, 19 | eqtr2i 2645 | 1 ⊢ (𝑂‘ 1 ) = 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 ‘cfv 5888 odcod 17944 1rcur 18501 chrcchr 19850 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-chr 19854 |
This theorem is referenced by: chrcl 19874 chrid 19875 chrdvds 19876 chrcong 19877 subrgchr 29794 ofldchr 29814 zrhchr 30020 |
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