Theorem ringrng 13883

Description: A unital ring is a non-unital ring. (Contributed by AV, 6-Jan-2020.)

Assertion

Ref	Expression
ringrng	|- ( R e. Ring -> R e. Rng )

Proof of Theorem ringrng

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ringabl 13879	. 2 |- ( R e. Ring -> R e. Abel )
2		eqid 2206	. . . 4 |- ( Base ` R ) = ( Base ` R )
3		eqid 2206	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
4		eqid 2206	. . . 4 |- ( +g ` R ) = ( +g ` R )
5		eqid 2206	. . . 4 |- ( .r ` R ) = ( .r ` R )
6	2, 3, 4, 5	isring 13847	. . 3 |- ( R e. Ring <-> ( R e. Grp /\ (mulGrp ` R ) e. Mnd /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) A. z e. ( Base ` R ) ( ( x ( .r ` R ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) /\ ( ( x ( +g ` R ) y ) ( .r ` R ) z ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) ) ) )
7		simpl 109	. . . . 5 |- ( ( R e. Abel /\ ( R e. Grp /\ (mulGrp ` R ) e. Mnd /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) A. z e. ( Base ` R ) ( ( x ( .r ` R ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) /\ ( ( x ( +g ` R ) y ) ( .r ` R ) z ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) ) ) ) -> R e. Abel )
8		mndsgrp 13338	. . . . . . 7 |- ( (mulGrp ` R ) e. Mnd -> (mulGrp ` R ) e. Smgrp )
9	8	3ad2ant2 1022	. . . . . 6 |- ( ( R e. Grp /\ (mulGrp ` R ) e. Mnd /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) A. z e. ( Base ` R ) ( ( x ( .r ` R ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) /\ ( ( x ( +g ` R ) y ) ( .r ` R ) z ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) ) ) -> (mulGrp ` R ) e. Smgrp )
10	9	adantl 277	. . . . 5 |- ( ( R e. Abel /\ ( R e. Grp /\ (mulGrp ` R ) e. Mnd /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) A. z e. ( Base ` R ) ( ( x ( .r ` R ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) /\ ( ( x ( +g ` R ) y ) ( .r ` R ) z ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) ) ) ) -> (mulGrp ` R ) e. Smgrp )
11		simpr3 1008	. . . . 5 |- ( ( R e. Abel /\ ( R e. Grp /\ (mulGrp ` R ) e. Mnd /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) A. z e. ( Base ` R ) ( ( x ( .r ` R ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) /\ ( ( x ( +g ` R ) y ) ( .r ` R ) z ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) ) ) ) -> A. x e. ( Base ` R ) A. y e. ( Base ` R ) A. z e. ( Base ` R ) ( ( x ( .r ` R ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) /\ ( ( x ( +g ` R ) y ) ( .r ` R ) z ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) ) )
12	2, 3, 4, 5	isrng 13781	. . . . 5 |- ( R e. Rng <-> ( R e. Abel /\ (mulGrp ` R ) e. Smgrp /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) A. z e. ( Base ` R ) ( ( x ( .r ` R ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) /\ ( ( x ( +g ` R ) y ) ( .r ` R ) z ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) ) ) )
13	7, 10, 11, 12	syl3anbrc 1184	. . . 4 |- ( ( R e. Abel /\ ( R e. Grp /\ (mulGrp ` R ) e. Mnd /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) A. z e. ( Base ` R ) ( ( x ( .r ` R ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) /\ ( ( x ( +g ` R ) y ) ( .r ` R ) z ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) ) ) ) -> R e. Rng )
14	13	ex 115	. . 3 |- ( R e. Abel -> ( ( R e. Grp /\ (mulGrp ` R ) e. Mnd /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) A. z e. ( Base ` R ) ( ( x ( .r ` R ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) /\ ( ( x ( +g ` R ) y ) ( .r ` R ) z ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) ) ) -> R e. Rng ) )
15	6, 14	biimtrid 152	. 2 |- ( R e. Abel -> ( R e. Ring -> R e. Rng ) )
16	1, 15	mpcom 36	1 |- ( R e. Ring -> R e. Rng )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2177   A.wral 2485   ` cfv 5285  (class class class)co 5962   Basecbs 12917   +g cplusg 12994   .rcmulr 12995  Smgrpcsgrp 13318   Mndcmnd 13333   Grpcgrp 13417   Abelcabl 13706  mulGrpcmgp 13767  Rngcrng 13779   Ringcrg 13843

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-addcom 8055  ax-addass 8057  ax-i2m1 8060  ax-0lt1 8061  ax-0id 8063  ax-rnegex 8064  ax-pre-ltirr 8067  ax-pre-ltadd 8071

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-pnf 8139  df-mnf 8140  df-ltxr 8142  df-inn 9067  df-2 9125  df-3 9126  df-ndx 12920  df-slot 12921  df-base 12923  df-sets 12924  df-plusg 13007  df-mulr 13008  df-0g 13175  df-mgm 13273  df-sgrp 13319  df-mnd 13334  df-grp 13420  df-minusg 13421  df-cmn 13707  df-abl 13708  df-mgp 13768  df-rng 13780  df-ur 13807  df-ring 13845

This theorem is referenced by:  ringssrng  13884  dflidl2  14335  df2idl2  14356  2idlcpbl  14371  quscrng  14380