![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > cnsubmlem | GIF version |
Description: Lemma for nn0subm 13853 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 3174 | . 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 2576 | . 2 β’ βπ₯ β π΄ βπ¦ β π΄ (π₯ + π¦) β π΄ |
6 | cnring 13840 | . . 3 β’ βfld β Ring | |
7 | ringmnd 13327 | . . 3 β’ (βfld β Ring β βfld β Mnd) | |
8 | cnfldbas 13835 | . . . 4 β’ β = (Baseββfld) | |
9 | cnfld0 13841 | . . . 4 β’ 0 = (0gββfld) | |
10 | cnfldadd 13836 | . . . 4 β’ + = (+gββfld) | |
11 | 8, 9, 10 | issubm 12896 | . . 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 1181 | 1 β’ π΄ β (SubMndββfld) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 β wb 105 β§ w3a 980 β wcel 2160 βwral 2468 β wss 3144 βcfv 5231 (class class class)co 5891 βcc 7828 0cc0 7830 + caddc 7833 Mndcmnd 12849 SubMndcsubmnd 12882 Ringcrg 13317 βfldccnfld 13831 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7921 ax-resscn 7922 ax-1cn 7923 ax-1re 7924 ax-icn 7925 ax-addcl 7926 ax-addrcl 7927 ax-mulcl 7928 ax-mulrcl 7929 ax-addcom 7930 ax-mulcom 7931 ax-addass 7932 ax-mulass 7933 ax-distr 7934 ax-i2m1 7935 ax-0lt1 7936 ax-1rid 7937 ax-0id 7938 ax-rnegex 7939 ax-precex 7940 ax-cnre 7941 ax-pre-ltirr 7942 ax-pre-ltwlin 7943 ax-pre-lttrn 7944 ax-pre-apti 7945 ax-pre-ltadd 7946 ax-pre-mulgt0 7947 ax-addf 7952 ax-mulf 7953 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-tp 3615 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-pnf 8013 df-mnf 8014 df-xr 8015 df-ltxr 8016 df-le 8017 df-sub 8149 df-neg 8150 df-reap 8551 df-inn 8939 df-2 8997 df-3 8998 df-4 8999 df-5 9000 df-6 9001 df-7 9002 df-8 9003 df-9 9004 df-n0 9196 df-z 9273 df-dec 9404 df-uz 9548 df-fz 10028 df-cj 10870 df-struct 12488 df-ndx 12489 df-slot 12490 df-base 12492 df-sets 12493 df-plusg 12574 df-mulr 12575 df-starv 12576 df-0g 12735 df-mgm 12804 df-sgrp 12837 df-mnd 12850 df-submnd 12884 df-grp 12920 df-cmn 13192 df-mgp 13242 df-ring 13319 df-cring 13320 df-icnfld 13832 |
This theorem is referenced by: nn0subm 13853 rege0subm 13854 |
Copyright terms: Public domain | W3C validator |