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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 20318 | . . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅)) | |
2 | eqid 2731 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
3 | 2 | subrg1cl 20320 | . . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑅) ∈ 𝐴) |
4 | eqid 2731 | . . . . 5 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴) | |
5 | eqid 2731 | . . . . 5 ⊢ (od‘𝑅) = (od‘𝑅) | |
6 | eqid 2731 | . . . . 5 ⊢ (od‘(𝑅 ↾s 𝐴)) = (od‘(𝑅 ↾s 𝐴)) | |
7 | 4, 5, 6 | subgod 19402 | . . . 4 ⊢ ((𝐴 ∈ (SubGrp‘𝑅) ∧ (1r‘𝑅) ∈ 𝐴) → ((od‘𝑅)‘(1r‘𝑅)) = ((od‘(𝑅 ↾s 𝐴))‘(1r‘𝑅))) |
8 | 1, 3, 7 | syl2anc 584 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((od‘𝑅)‘(1r‘𝑅)) = ((od‘(𝑅 ↾s 𝐴))‘(1r‘𝑅))) |
9 | 4, 2 | subrg1 20322 | . . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (1r‘𝑅) = (1r‘(𝑅 ↾s 𝐴))) |
10 | 9 | fveq2d 6882 | . . 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 2731 | . . 3 ⊢ (1r‘(𝑅 ↾s 𝐴)) = (1r‘(𝑅 ↾s 𝐴)) | |
13 | eqid 2731 | . . 3 ⊢ (chr‘(𝑅 ↾s 𝐴)) = (chr‘(𝑅 ↾s 𝐴)) | |
14 | 6, 12, 13 | chrval 21010 | . 2 ⊢ ((od‘(𝑅 ↾s 𝐴))‘(1r‘(𝑅 ↾s 𝐴))) = (chr‘(𝑅 ↾s 𝐴)) |
15 | eqid 2731 | . . 3 ⊢ (chr‘𝑅) = (chr‘𝑅) | |
16 | 5, 2, 15 | chrval 21010 | . 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 1541 ∈ wcel 2106 ‘cfv 6532 (class class class)co 7393 ↾s cress 17155 SubGrpcsubg 18972 odcod 19356 1rcur 19963 SubRingcsubrg 20308 chrcchr 20984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-sup 9419 df-inf 9420 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-n0 12455 df-z 12541 df-uz 12805 df-seq 13949 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-0g 17369 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-submnd 18648 df-grp 18797 df-minusg 18798 df-mulg 18923 df-subg 18975 df-od 19360 df-mgp 19947 df-ur 19964 df-ring 20016 df-subrg 20310 df-chr 20988 |
This theorem is referenced by: primefldchr 32261 fldextchr 32582 cnrrext 32821 |
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