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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 6665 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (od‘𝑟) = (od‘𝑅)) | |
3 | chrval.o | . . . . . 6 ⊢ 𝑂 = (od‘𝑅) | |
4 | 2, 3 | syl6eqr 2874 | . . . . 5 ⊢ (𝑟 = 𝑅 → (od‘𝑟) = 𝑂) |
5 | fveq2 6665 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅)) | |
6 | chrval.u | . . . . . 6 ⊢ 1 = (1r‘𝑅) | |
7 | 5, 6 | syl6eqr 2874 | . . . . 5 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 ) |
8 | 4, 7 | fveq12d 6672 | . . . 4 ⊢ (𝑟 = 𝑅 → ((od‘𝑟)‘(1r‘𝑟)) = (𝑂‘ 1 )) |
9 | df-chr 20647 | . . . 4 ⊢ chr = (𝑟 ∈ V ↦ ((od‘𝑟)‘(1r‘𝑟))) | |
10 | fvex 6678 | . . . 4 ⊢ (𝑂‘ 1 ) ∈ V | |
11 | 8, 9, 10 | fvmpt 6763 | . . 3 ⊢ (𝑅 ∈ V → (chr‘𝑅) = (𝑂‘ 1 )) |
12 | fvprc 6658 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (chr‘𝑅) = ∅) | |
13 | fvprc 6658 | . . . . . . 7 ⊢ (¬ 𝑅 ∈ V → (od‘𝑅) = ∅) | |
14 | 3, 13 | syl5eq 2868 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ∅) |
15 | 14 | fveq1d 6667 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝑂‘ 1 ) = (∅‘ 1 )) |
16 | 0fv 6704 | . . . . 5 ⊢ (∅‘ 1 ) = ∅ | |
17 | 15, 16 | syl6eq 2872 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑂‘ 1 ) = ∅) |
18 | 12, 17 | eqtr4d 2859 | . . 3 ⊢ (¬ 𝑅 ∈ V → (chr‘𝑅) = (𝑂‘ 1 )) |
19 | 11, 18 | pm2.61i 184 | . 2 ⊢ (chr‘𝑅) = (𝑂‘ 1 ) |
20 | 1, 19 | eqtr2i 2845 | 1 ⊢ (𝑂‘ 1 ) = 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ∅c0 4291 ‘cfv 6350 odcod 18646 1rcur 19245 chrcchr 20643 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-iota 6309 df-fun 6352 df-fv 6358 df-chr 20647 |
This theorem is referenced by: chrcl 20667 chrid 20668 chrdvds 20669 chrcong 20670 dvdschrmulg 30853 subrgchr 30860 ofldchr 30882 zrhchr 31212 |
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