Theorem dmnnzd 33874

Description: A domain has no zero-divisors (besides zero). (Contributed by Jeff Madsen, 19-Jun-2010.)

Hypotheses

Ref	Expression
dmnnzd.1	|- G = ( 1st ` R )
dmnnzd.2	|- H = ( 2nd ` R )
dmnnzd.3	|- X = ran G
dmnnzd.4	|- Z = (GId ` G )

Assertion

Ref	Expression
dmnnzd	|- ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ ( A H B ) = Z ) ) -> ( A = Z \/ B = Z ) )

Proof of Theorem dmnnzd

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmnnzd.1	. . . . . 6 |- G = ( 1st ` R )
2		dmnnzd.2	. . . . . 6 |- H = ( 2nd ` R )
3		dmnnzd.3	. . . . . 6 |- X = ran G
4		dmnnzd.4	. . . . . 6 |- Z = (GId ` G )
5		eqid 2622	. . . . . 6 |- (GId ` H ) = (GId ` H )
6	1, 2, 3, 4, 5	isdmn3 33873	. . . . 5 |- ( R e. Dmn <-> ( R e. CRingOps /\ (GId ` H ) =/= Z /\ A. a e. X A. b e. X ( ( a H b ) = Z -> ( a = Z \/ b = Z ) ) ) )
7	6	simp3bi 1078	. . . 4 |- ( R e. Dmn -> A. a e. X A. b e. X ( ( a H b ) = Z -> ( a = Z \/ b = Z ) ) )
8		oveq1 6657	. . . . . . 7 |- ( a = A -> ( a H b ) = ( A H b ) )
9	8	eqeq1d 2624	. . . . . 6 |- ( a = A -> ( ( a H b ) = Z <-> ( A H b ) = Z ) )
10		eqeq1 2626	. . . . . . 7 |- ( a = A -> ( a = Z <-> A = Z ) )
11	10	orbi1d 739	. . . . . 6 |- ( a = A -> ( ( a = Z \/ b = Z ) <-> ( A = Z \/ b = Z ) ) )
12	9, 11	imbi12d 334	. . . . 5 |- ( a = A -> ( ( ( a H b ) = Z -> ( a = Z \/ b = Z ) ) <-> ( ( A H b ) = Z -> ( A = Z \/ b = Z ) ) ) )
13		oveq2 6658	. . . . . . 7 |- ( b = B -> ( A H b ) = ( A H B ) )
14	13	eqeq1d 2624	. . . . . 6 |- ( b = B -> ( ( A H b ) = Z <-> ( A H B ) = Z ) )
15		eqeq1 2626	. . . . . . 7 |- ( b = B -> ( b = Z <-> B = Z ) )
16	15	orbi2d 738	. . . . . 6 |- ( b = B -> ( ( A = Z \/ b = Z ) <-> ( A = Z \/ B = Z ) ) )
17	14, 16	imbi12d 334	. . . . 5 |- ( b = B -> ( ( ( A H b ) = Z -> ( A = Z \/ b = Z ) ) <-> ( ( A H B ) = Z -> ( A = Z \/ B = Z ) ) ) )
18	12, 17	rspc2v 3322	. . . 4 |- ( ( A e. X /\ B e. X ) -> ( A. a e. X A. b e. X ( ( a H b ) = Z -> ( a = Z \/ b = Z ) ) -> ( ( A H B ) = Z -> ( A = Z \/ B = Z ) ) ) )
19	7, 18	syl5com 31	. . 3 |- ( R e. Dmn -> ( ( A e. X /\ B e. X ) -> ( ( A H B ) = Z -> ( A = Z \/ B = Z ) ) ) )
20	19	expd 452	. 2 |- ( R e. Dmn -> ( A e. X -> ( B e. X -> ( ( A H B ) = Z -> ( A = Z \/ B = Z ) ) ) ) )
21	20	3imp2 1282	1 |- ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ ( A H B ) = Z ) ) -> ( A = Z \/ B = Z ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167  GIdcgi 27344  CRingOpsccring 33792   Dmncdmn 33846

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-grpo 27347  df-gid 27348  df-ginv 27349  df-ablo 27399  df-ass 33642  df-exid 33644  df-mgmOLD 33648  df-sgrOLD 33660  df-mndo 33666  df-rngo 33694  df-com2 33789  df-crngo 33793  df-idl 33809  df-pridl 33810  df-prrngo 33847  df-dmn 33848  df-igen 33859

This theorem is referenced by:  dmncan1  33875