Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > subrgchr | Structured version Visualization version GIF version |
Description: If 𝐴 is a subring of 𝑅, then they have the same characteristic. (Contributed by Thierry Arnoux, 24-Feb-2018.) |
Ref | Expression |
---|---|
subrgchr | ⊢ (𝐴 ∈ (SubRing‘𝑅) → (chr‘(𝑅 ↾s 𝐴)) = (chr‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subrgsubg 19760 | . . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅)) | |
2 | eqid 2736 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
3 | 2 | subrg1cl 19762 | . . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑅) ∈ 𝐴) |
4 | eqid 2736 | . . . . 5 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴) | |
5 | eqid 2736 | . . . . 5 ⊢ (od‘𝑅) = (od‘𝑅) | |
6 | eqid 2736 | . . . . 5 ⊢ (od‘(𝑅 ↾s 𝐴)) = (od‘(𝑅 ↾s 𝐴)) | |
7 | 4, 5, 6 | subgod 18913 | . . . 4 ⊢ ((𝐴 ∈ (SubGrp‘𝑅) ∧ (1r‘𝑅) ∈ 𝐴) → ((od‘𝑅)‘(1r‘𝑅)) = ((od‘(𝑅 ↾s 𝐴))‘(1r‘𝑅))) |
8 | 1, 3, 7 | syl2anc 587 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((od‘𝑅)‘(1r‘𝑅)) = ((od‘(𝑅 ↾s 𝐴))‘(1r‘𝑅))) |
9 | 4, 2 | subrg1 19764 | . . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑅) = (1r‘(𝑅 ↾s 𝐴))) |
10 | 9 | fveq2d 6699 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((od‘(𝑅 ↾s 𝐴))‘(1r‘𝑅)) = ((od‘(𝑅 ↾s 𝐴))‘(1r‘(𝑅 ↾s 𝐴)))) |
11 | 8, 10 | eqtr2d 2772 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((od‘(𝑅 ↾s 𝐴))‘(1r‘(𝑅 ↾s 𝐴))) = ((od‘𝑅)‘(1r‘𝑅))) |
12 | eqid 2736 | . . 3 ⊢ (1r‘(𝑅 ↾s 𝐴)) = (1r‘(𝑅 ↾s 𝐴)) | |
13 | eqid 2736 | . . 3 ⊢ (chr‘(𝑅 ↾s 𝐴)) = (chr‘(𝑅 ↾s 𝐴)) | |
14 | 6, 12, 13 | chrval 20444 | . 2 ⊢ ((od‘(𝑅 ↾s 𝐴))‘(1r‘(𝑅 ↾s 𝐴))) = (chr‘(𝑅 ↾s 𝐴)) |
15 | eqid 2736 | . . 3 ⊢ (chr‘𝑅) = (chr‘𝑅) | |
16 | 5, 2, 15 | chrval 20444 | . 2 ⊢ ((od‘𝑅)‘(1r‘𝑅)) = (chr‘𝑅) |
17 | 11, 14, 16 | 3eqtr3g 2794 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (chr‘(𝑅 ↾s 𝐴)) = (chr‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ‘cfv 6358 (class class class)co 7191 ↾s cress 16667 SubGrpcsubg 18491 odcod 18870 1rcur 19470 SubRingcsubrg 19750 chrcchr 20422 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-sup 9036 df-inf 9037 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-n0 12056 df-z 12142 df-uz 12404 df-seq 13540 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-0g 16900 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-submnd 18173 df-grp 18322 df-minusg 18323 df-mulg 18443 df-subg 18494 df-od 18874 df-mgp 19459 df-ur 19471 df-ring 19518 df-subrg 19752 df-chr 20426 |
This theorem is referenced by: primefldchr 31166 fldextchr 31408 cnrrext 31626 |
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