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Mirrors > Home > ILE Home > Th. List > cnsubmlem | GIF version |
Description: Lemma for nn0subm 13734 and friends. (Contributed by Mario Carneiro, 18-Jun-2015.) |
Ref | Expression |
---|---|
cnsubglem.1 | β’ (π₯ β π΄ β π₯ β β) |
cnsubglem.2 | β’ ((π₯ β π΄ β§ π¦ β π΄) β (π₯ + π¦) β π΄) |
cnsubmlem.3 | β’ 0 β π΄ |
Ref | Expression |
---|---|
cnsubmlem | β’ π΄ β (SubMndββfld) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnsubglem.1 | . . 3 β’ (π₯ β π΄ β π₯ β β) | |
2 | 1 | ssriv 3171 | . 2 β’ π΄ β β |
3 | cnsubmlem.3 | . 2 β’ 0 β π΄ | |
4 | cnsubglem.2 | . . 3 β’ ((π₯ β π΄ β§ π¦ β π΄) β (π₯ + π¦) β π΄) | |
5 | 4 | rgen2 2573 | . 2 β’ βπ₯ β π΄ βπ¦ β π΄ (π₯ + π¦) β π΄ |
6 | cnring 13721 | . . 3 β’ βfld β Ring | |
7 | ringmnd 13253 | . . 3 β’ (βfld β Ring β βfld β Mnd) | |
8 | cnfldbas 13716 | . . . 4 β’ β = (Baseββfld) | |
9 | cnfld0 13722 | . . . 4 β’ 0 = (0gββfld) | |
10 | cnfldadd 13717 | . . . 4 β’ + = (+gββfld) | |
11 | 8, 9, 10 | issubm 12884 | . . 3 β’ (βfld β Mnd β (π΄ β (SubMndββfld) β (π΄ β β β§ 0 β π΄ β§ βπ₯ β π΄ βπ¦ β π΄ (π₯ + π¦) β π΄))) |
12 | 6, 7, 11 | mp2b 8 | . 2 β’ (π΄ β (SubMndββfld) β (π΄ β β β§ 0 β π΄ β§ βπ₯ β π΄ βπ¦ β π΄ (π₯ + π¦) β π΄)) |
13 | 2, 3, 5, 12 | mpbir3an 1180 | 1 β’ π΄ β (SubMndββfld) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 β wb 105 β§ w3a 979 β wcel 2158 βwral 2465 β wss 3141 βcfv 5228 (class class class)co 5888 βcc 7822 0cc0 7824 + caddc 7827 Mndcmnd 12838 SubMndcsubmnd 12871 Ringcrg 13243 βfldccnfld 13712 |
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 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 ax-addf 7946 ax-mulf 7947 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-tp 3612 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-reap 8545 df-inn 8933 df-2 8991 df-3 8992 df-4 8993 df-5 8994 df-6 8995 df-7 8996 df-8 8997 df-9 8998 df-n0 9190 df-z 9267 df-dec 9398 df-uz 9542 df-fz 10022 df-cj 10864 df-struct 12477 df-ndx 12478 df-slot 12479 df-base 12481 df-sets 12482 df-plusg 12563 df-mulr 12564 df-starv 12565 df-0g 12724 df-mgm 12793 df-sgrp 12826 df-mnd 12839 df-submnd 12873 df-grp 12901 df-cmn 13122 df-mgp 13171 df-ring 13245 df-cring 13246 df-icnfld 13713 |
This theorem is referenced by: nn0subm 13734 rege0subm 13735 |
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