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Mirrors > Home > MPE Home > Th. List > subrgmvr | Unicode version |
Description: The variables in a subring polynomial algebra are the same as the original ring. (Contributed by Mario Carneiro, 4-Jul-2015.) |
Ref | Expression |
---|---|
subrgmvr.v |
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subrgmvr.i |
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subrgmvr.r |
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subrgmvr.h |
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Ref | Expression |
---|---|
subrgmvr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subrgmvr.r |
. . . . . 6
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2 | subrgmvr.h |
. . . . . . 7
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3 | eqid 2408 |
. . . . . . 7
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4 | 2, 3 | subrg1 15837 |
. . . . . 6
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5 | 1, 4 | syl 16 |
. . . . 5
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6 | eqid 2408 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 2, 6 | subrg0 15834 |
. . . . . 6
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8 | 1, 7 | syl 16 |
. . . . 5
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9 | 5, 8 | ifeq12d 3719 |
. . . 4
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10 | 9 | mpteq2dv 4260 |
. . 3
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11 | 10 | mpteq2dv 4260 |
. 2
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12 | subrgmvr.v |
. . 3
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13 | eqid 2408 |
. . 3
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14 | subrgmvr.i |
. . 3
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15 | subrgrcl 15832 |
. . . 4
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16 | 1, 15 | syl 16 |
. . 3
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17 | 12, 13, 6, 3, 14, 16 | mvrfval 16443 |
. 2
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18 | eqid 2408 |
. . 3
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19 | eqid 2408 |
. . 3
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20 | eqid 2408 |
. . 3
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21 | ovex 6069 |
. . . . 5
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22 | 2, 21 | eqeltri 2478 |
. . . 4
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23 | 22 | a1i 11 |
. . 3
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24 | 18, 13, 19, 20, 14, 23 | mvrfval 16443 |
. 2
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25 | 11, 17, 24 | 3eqtr4d 2450 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: subrgmvrf 16484 subrgvr1 16613 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2389 ax-rep 4284 ax-sep 4294 ax-nul 4302 ax-pow 4341 ax-pr 4367 ax-un 4664 ax-cnex 9006 ax-resscn 9007 ax-1cn 9008 ax-icn 9009 ax-addcl 9010 ax-addrcl 9011 ax-mulcl 9012 ax-mulrcl 9013 ax-mulcom 9014 ax-addass 9015 ax-mulass 9016 ax-distr 9017 ax-i2m1 9018 ax-1ne0 9019 ax-1rid 9020 ax-rnegex 9021 ax-rrecex 9022 ax-cnre 9023 ax-pre-lttri 9024 ax-pre-lttrn 9025 ax-pre-ltadd 9026 ax-pre-mulgt0 9027 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3or 937 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2262 df-mo 2263 df-clab 2395 df-cleq 2401 df-clel 2404 df-nfc 2533 df-ne 2573 df-nel 2574 df-ral 2675 df-rex 2676 df-reu 2677 df-rmo 2678 df-rab 2679 df-v 2922 df-sbc 3126 df-csb 3216 df-dif 3287 df-un 3289 df-in 3291 df-ss 3298 df-pss 3300 df-nul 3593 df-if 3704 df-pw 3765 df-sn 3784 df-pr 3785 df-tp 3786 df-op 3787 df-uni 3980 df-iun 4059 df-br 4177 df-opab 4231 df-mpt 4232 df-tr 4267 df-eprel 4458 df-id 4462 df-po 4467 df-so 4468 df-fr 4505 df-we 4507 df-ord 4548 df-on 4549 df-lim 4550 df-suc 4551 df-om 4809 df-xp 4847 df-rel 4848 df-cnv 4849 df-co 4850 df-dm 4851 df-rn 4852 df-res 4853 df-ima 4854 df-iota 5381 df-fun 5419 df-fn 5420 df-f 5421 df-f1 5422 df-fo 5423 df-f1o 5424 df-fv 5425 df-ov 6047 df-oprab 6048 df-mpt2 6049 df-riota 6512 df-recs 6596 df-rdg 6631 df-er 6868 df-en 7073 df-dom 7074 df-sdom 7075 df-pnf 9082 df-mnf 9083 df-xr 9084 df-ltxr 9085 df-le 9086 df-sub 9253 df-neg 9254 df-nn 9961 df-2 10018 df-3 10019 df-ndx 13431 df-slot 13432 df-base 13433 df-sets 13434 df-ress 13435 df-plusg 13501 df-mulr 13502 df-0g 13686 df-mnd 14649 df-grp 14771 df-subg 14900 df-mgp 15608 df-rng 15622 df-ur 15624 df-subrg 15825 df-mvr 16377 |
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