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| 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 6906 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (od‘𝑟) = (od‘𝑅)) | |
| 3 | chrval.o | . . . . . 6 ⊢ 𝑂 = (od‘𝑅) | |
| 4 | 2, 3 | eqtr4di 2795 | . . . . 5 ⊢ (𝑟 = 𝑅 → (od‘𝑟) = 𝑂) | 
| 5 | fveq2 6906 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅)) | |
| 6 | chrval.u | . . . . . 6 ⊢ 1 = (1r‘𝑅) | |
| 7 | 5, 6 | eqtr4di 2795 | . . . . 5 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 ) | 
| 8 | 4, 7 | fveq12d 6913 | . . . 4 ⊢ (𝑟 = 𝑅 → ((od‘𝑟)‘(1r‘𝑟)) = (𝑂‘ 1 )) | 
| 9 | df-chr 21516 | . . . 4 ⊢ chr = (𝑟 ∈ V ↦ ((od‘𝑟)‘(1r‘𝑟))) | |
| 10 | fvex 6919 | . . . 4 ⊢ (𝑂‘ 1 ) ∈ V | |
| 11 | 8, 9, 10 | fvmpt 7016 | . . 3 ⊢ (𝑅 ∈ V → (chr‘𝑅) = (𝑂‘ 1 )) | 
| 12 | fvprc 6898 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (chr‘𝑅) = ∅) | |
| 13 | fvprc 6898 | . . . . . . 7 ⊢ (¬ 𝑅 ∈ V → (od‘𝑅) = ∅) | |
| 14 | 3, 13 | eqtrid 2789 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ∅) | 
| 15 | 14 | fveq1d 6908 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝑂‘ 1 ) = (∅‘ 1 )) | 
| 16 | 0fv 6950 | . . . . 5 ⊢ (∅‘ 1 ) = ∅ | |
| 17 | 15, 16 | eqtrdi 2793 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑂‘ 1 ) = ∅) | 
| 18 | 12, 17 | eqtr4d 2780 | . . 3 ⊢ (¬ 𝑅 ∈ V → (chr‘𝑅) = (𝑂‘ 1 )) | 
| 19 | 11, 18 | pm2.61i 182 | . 2 ⊢ (chr‘𝑅) = (𝑂‘ 1 ) | 
| 20 | 1, 19 | eqtr2i 2766 | 1 ⊢ (𝑂‘ 1 ) = 𝐶 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 ‘cfv 6561 odcod 19542 1rcur 20178 chrcchr 21512 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-chr 21516 | 
| This theorem is referenced by: chrcl 21539 chrid 21540 chrdvds 21541 chrcong 21542 dvdschrmulg 21543 ply1chr 22310 subrgchr 33241 ofldchr 33344 zrhchr 33975 | 
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