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Mirrors > Home > MPE Home > Th. List > subrg1 | Structured version Unicode version |
Description: A subring always has the same multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
Ref | Expression |
---|---|
subrg1.1 |
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subrg1.2 |
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Ref | Expression |
---|---|
subrg1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2450 |
. . . . 5
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2 | 1 | subrg1cl 16965 |
. . . 4
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3 | subrg1.1 |
. . . . 5
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4 | 3 | subrgbas 16966 |
. . . 4
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5 | 2, 4 | eleqtrd 2538 |
. . 3
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6 | eqid 2450 |
. . . . . . . 8
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7 | 6 | subrgss 16958 |
. . . . . . 7
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8 | 4, 7 | eqsstr3d 3475 |
. . . . . 6
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9 | 8 | sselda 3440 |
. . . . 5
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10 | subrgrcl 16962 |
. . . . . . 7
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11 | eqid 2450 |
. . . . . . . 8
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12 | 6, 11, 1 | rngidmlem 16759 |
. . . . . . 7
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13 | 10, 12 | sylan 471 |
. . . . . 6
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14 | 3, 11 | ressmulr 14379 |
. . . . . . . . . 10
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15 | 14 | oveqd 6193 |
. . . . . . . . 9
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16 | 15 | eqeq1d 2452 |
. . . . . . . 8
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17 | 14 | oveqd 6193 |
. . . . . . . . 9
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18 | 17 | eqeq1d 2452 |
. . . . . . . 8
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19 | 16, 18 | anbi12d 710 |
. . . . . . 7
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20 | 19 | biimpa 484 |
. . . . . 6
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21 | 13, 20 | syldan 470 |
. . . . 5
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22 | 9, 21 | syldan 470 |
. . . 4
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23 | 22 | ralrimiva 2881 |
. . 3
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24 | 3 | subrgrng 16960 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | eqid 2450 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
26 | eqid 2450 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | eqid 2450 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
28 | 25, 26, 27 | isrngid 16762 |
. . . 4
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29 | 24, 28 | syl 16 |
. . 3
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30 | 5, 23, 29 | mpbi2and 912 |
. 2
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31 | subrg1.2 |
. 2
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32 | 30, 31 | syl6reqr 2509 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1709 ax-7 1729 ax-8 1759 ax-9 1761 ax-10 1776 ax-11 1781 ax-12 1793 ax-13 1944 ax-ext 2429 ax-sep 4497 ax-nul 4505 ax-pow 4554 ax-pr 4615 ax-un 6458 ax-cnex 9425 ax-resscn 9426 ax-1cn 9427 ax-icn 9428 ax-addcl 9429 ax-addrcl 9430 ax-mulcl 9431 ax-mulrcl 9432 ax-mulcom 9433 ax-addass 9434 ax-mulass 9435 ax-distr 9436 ax-i2m1 9437 ax-1ne0 9438 ax-1rid 9439 ax-rnegex 9440 ax-rrecex 9441 ax-cnre 9442 ax-pre-lttri 9443 ax-pre-lttrn 9444 ax-pre-ltadd 9445 ax-pre-mulgt0 9446 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1702 df-eu 2263 df-mo 2264 df-clab 2436 df-cleq 2442 df-clel 2445 df-nfc 2598 df-ne 2643 df-nel 2644 df-ral 2797 df-rex 2798 df-reu 2799 df-rmo 2800 df-rab 2801 df-v 3056 df-sbc 3271 df-csb 3373 df-dif 3415 df-un 3417 df-in 3419 df-ss 3426 df-pss 3428 df-nul 3722 df-if 3876 df-pw 3946 df-sn 3962 df-pr 3964 df-tp 3966 df-op 3968 df-uni 4176 df-iun 4257 df-br 4377 df-opab 4435 df-mpt 4436 df-tr 4470 df-eprel 4716 df-id 4720 df-po 4725 df-so 4726 df-fr 4763 df-we 4765 df-ord 4806 df-on 4807 df-lim 4808 df-suc 4809 df-xp 4930 df-rel 4931 df-cnv 4932 df-co 4933 df-dm 4934 df-rn 4935 df-res 4936 df-ima 4937 df-iota 5465 df-fun 5504 df-fn 5505 df-f 5506 df-f1 5507 df-fo 5508 df-f1o 5509 df-fv 5510 df-riota 6137 df-ov 6179 df-oprab 6180 df-mpt2 6181 df-om 6563 df-recs 6918 df-rdg 6952 df-er 7187 df-en 7397 df-dom 7398 df-sdom 7399 df-pnf 9507 df-mnf 9508 df-xr 9509 df-ltxr 9510 df-le 9511 df-sub 9684 df-neg 9685 df-nn 10410 df-2 10467 df-3 10468 df-ndx 14265 df-slot 14266 df-base 14267 df-sets 14268 df-ress 14269 df-plusg 14339 df-mulr 14340 df-0g 14468 df-mnd 15503 df-subg 15766 df-mgp 16683 df-ur 16695 df-rng 16739 df-subrg 16955 |
This theorem is referenced by: subrguss 16972 subrginv 16973 subrgunit 16975 subsubrg 16983 sralmod 17360 subrgnzr 17441 ressascl 17506 mpl1 17616 subrgmvr 17633 gzrngunitlem 17972 zring1 17989 zrng1 17995 prmirredlemOLD 18015 mulgrhmOLD 18024 mulgrhm2OLD 18025 re1r 18138 clm1 20747 qrng1 22973 subrgchr 26382 scmatsrng1 31030 |
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