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| Mirrors > Home > ILE Home > Th. List > subrgintm | Unicode version | ||
| Description: The intersection of an inhabited collection of subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.) |
| Ref | Expression |
|---|---|
| subrgintm |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | subrgsubg 14022 |
. . . . 5
| |
| 2 | 1 | ssriv 3197 |
. . . 4
|
| 3 | sstr 3201 |
. . . 4
| |
| 4 | 2, 3 | mpan2 425 |
. . 3
|
| 5 | subgintm 13567 |
. . 3
| |
| 6 | 4, 5 | sylan 283 |
. 2
|
| 7 | ssel2 3188 |
. . . . . 6
| |
| 8 | 7 | adantlr 477 |
. . . . 5
|
| 9 | eqid 2205 |
. . . . . 6
| |
| 10 | 9 | subrg1cl 14024 |
. . . . 5
|
| 11 | 8, 10 | syl 14 |
. . . 4
|
| 12 | 11 | ralrimiva 2579 |
. . 3
|
| 13 | ssel 3187 |
. . . . . . 7
| |
| 14 | subrgrcl 14021 |
. . . . . . 7
| |
| 15 | 13, 14 | syl6 33 |
. . . . . 6
|
| 16 | 15 | exlimdv 1842 |
. . . . 5
|
| 17 | 16 | imp 124 |
. . . 4
|
| 18 | ringsrg 13842 |
. . . 4
| |
| 19 | eqid 2205 |
. . . . . 6
| |
| 20 | 19, 9 | srgidcl 13771 |
. . . . 5
|
| 21 | elintg 3893 |
. . . . 5
| |
| 22 | 20, 21 | syl 14 |
. . . 4
|
| 23 | 17, 18, 22 | 3syl 17 |
. . 3
|
| 24 | 12, 23 | mpbird 167 |
. 2
|
| 25 | 8 | adantlr 477 |
. . . . . 6
|
| 26 | simprl 529 |
. . . . . . 7
| |
| 27 | elinti 3894 |
. . . . . . . 8
| |
| 28 | 27 | imp 124 |
. . . . . . 7
|
| 29 | 26, 28 | sylan 283 |
. . . . . 6
|
| 30 | simprr 531 |
. . . . . . 7
| |
| 31 | elinti 3894 |
. . . . . . . 8
| |
| 32 | 31 | imp 124 |
. . . . . . 7
|
| 33 | 30, 32 | sylan 283 |
. . . . . 6
|
| 34 | eqid 2205 |
. . . . . . 7
| |
| 35 | 34 | subrgmcl 14028 |
. . . . . 6
|
| 36 | 25, 29, 33, 35 | syl3anc 1250 |
. . . . 5
|
| 37 | 36 | ralrimiva 2579 |
. . . 4
|
| 38 | simplr 528 |
. . . . . 6
| |
| 39 | eleq1w 2266 |
. . . . . . . 8
| |
| 40 | 39 | cbvexv 1942 |
. . . . . . 7
|
| 41 | 36 | elexd 2785 |
. . . . . . . . 9
|
| 42 | 41 | ex 115 |
. . . . . . . 8
|
| 43 | 42 | exlimdv 1842 |
. . . . . . 7
|
| 44 | 40, 43 | biimtrrid 153 |
. . . . . 6
|
| 45 | 38, 44 | mpd 13 |
. . . . 5
|
| 46 | elintg 3893 |
. . . . 5
| |
| 47 | 45, 46 | syl 14 |
. . . 4
|
| 48 | 37, 47 | mpbird 167 |
. . 3
|
| 49 | 48 | ralrimivva 2588 |
. 2
|
| 50 | 19, 9, 34 | issubrg2 14036 |
. . 3
|
| 51 | 17, 50 | syl 14 |
. 2
|
| 52 | 6, 24, 49, 51 | mpbir3and 1183 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4160 ax-sep 4163 ax-pow 4219 ax-pr 4254 ax-un 4481 ax-setind 4586 ax-cnex 8018 ax-resscn 8019 ax-1cn 8020 ax-1re 8021 ax-icn 8022 ax-addcl 8023 ax-addrcl 8024 ax-mulcl 8025 ax-addcom 8027 ax-addass 8029 ax-i2m1 8032 ax-0lt1 8033 ax-0id 8035 ax-rnegex 8036 ax-pre-ltirr 8039 ax-pre-lttrn 8041 ax-pre-ltadd 8043 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4046 df-opab 4107 df-mpt 4108 df-id 4341 df-xp 4682 df-rel 4683 df-cnv 4684 df-co 4685 df-dm 4686 df-rn 4687 df-res 4688 df-ima 4689 df-iota 5233 df-fun 5274 df-fn 5275 df-f 5276 df-f1 5277 df-fo 5278 df-f1o 5279 df-fv 5280 df-riota 5901 df-ov 5949 df-oprab 5950 df-mpo 5951 df-pnf 8111 df-mnf 8112 df-ltxr 8114 df-inn 9039 df-2 9097 df-3 9098 df-ndx 12868 df-slot 12869 df-base 12871 df-sets 12872 df-iress 12873 df-plusg 12955 df-mulr 12956 df-0g 13123 df-mgm 13221 df-sgrp 13267 df-mnd 13282 df-grp 13368 df-minusg 13369 df-subg 13539 df-cmn 13655 df-abl 13656 df-mgp 13716 df-ur 13755 df-srg 13759 df-ring 13793 df-subrg 14014 |
| This theorem is referenced by: subrgin 14039 |
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