Theorem unxpwdom3 37665

Description: Weaker version of unxpwdom 8494 where a function is required only to be cancellative, not an injection. D and B are to be thought of as "large" "horizonal" sets, the others as "small". Because the operator is row-wise injective, but the whole row cannot inject into A, each row must hit an element of B; by column injectivity, each row can be identified in at least one way by the B element that it hits and the column in which it is hit. (Contributed by Stefan O'Rear, 8-Jul-2015.) MOVABLE

Hypotheses

Ref	Expression
unxpwdom3.av	|- ( ph -> A e. V )
unxpwdom3.bv	|- ( ph -> B e. W )
unxpwdom3.dv	|- ( ph -> D e. X )
unxpwdom3.ov	|- ( ( ph /\ a e. C /\ b e. D ) -> ( a .+ b ) e. ( A u. B ) )
unxpwdom3.lc	|- ( ( ( ph /\ a e. C ) /\ ( b e. D /\ c e. D ) ) -> ( ( a .+ b ) = ( a .+ c ) <-> b = c ) )
unxpwdom3.rc	|- ( ( ( ph /\ d e. D ) /\ ( a e. C /\ c e. C ) ) -> ( ( c .+ d ) = ( a .+ d ) <-> c = a ) )
unxpwdom3.ni	|- ( ph -> -. D ~<_ A )

Assertion

Ref	Expression
unxpwdom3	|- ( ph -> C ~<_* ( D X. B ) )

Distinct variable groups:   a, b, c, d, B   C, a, b, c, d   D, a, b, c, d   .+ , a, b, c, d   ph, a, b, c, d   A, b, c

Allowed substitution hints:   A( a, d)   V( a, b, c, d)   W( a, b, c, d)   X( a, b, c, d)

Proof of Theorem unxpwdom3

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

Step	Hyp	Ref	Expression
1		unxpwdom3.dv	. . 3 |- ( ph -> D e. X )
2		unxpwdom3.bv	. . 3 |- ( ph -> B e. W )
3		xpexg 6960	. . 3 |- ( ( D e. X /\ B e. W ) -> ( D X. B ) e. _V )
4	1, 2, 3	syl2anc 693	. 2 |- ( ph -> ( D X. B ) e. _V )
5		simprr 796	. . . . 5 |- ( ( ( ph /\ a e. C ) /\ ( d e. D /\ ( a .+ d ) e. B ) ) -> ( a .+ d ) e. B )
6		simplr 792	. . . . . . 7 |- ( ( ( ph /\ a e. C ) /\ ( d e. D /\ ( a .+ d ) e. B ) ) -> a e. C )
7		unxpwdom3.rc	. . . . . . . . . 10 |- ( ( ( ph /\ d e. D ) /\ ( a e. C /\ c e. C ) ) -> ( ( c .+ d ) = ( a .+ d ) <-> c = a ) )
8	7	an4s 869	. . . . . . . . 9 |- ( ( ( ph /\ a e. C ) /\ ( d e. D /\ c e. C ) ) -> ( ( c .+ d ) = ( a .+ d ) <-> c = a ) )
9	8	anassrs 680	. . . . . . . 8 |- ( ( ( ( ph /\ a e. C ) /\ d e. D ) /\ c e. C ) -> ( ( c .+ d ) = ( a .+ d ) <-> c = a ) )
10	9	adantlrr 757	. . . . . . 7 |- ( ( ( ( ph /\ a e. C ) /\ ( d e. D /\ ( a .+ d ) e. B ) ) /\ c e. C ) -> ( ( c .+ d ) = ( a .+ d ) <-> c = a ) )
11	6, 10	riota5 6637	. . . . . 6 |- ( ( ( ph /\ a e. C ) /\ ( d e. D /\ ( a .+ d ) e. B ) ) -> ( iota_ c e. C ( c .+ d ) = ( a .+ d ) ) = a )
12	11	eqcomd 2628	. . . . 5 |- ( ( ( ph /\ a e. C ) /\ ( d e. D /\ ( a .+ d ) e. B ) ) -> a = ( iota_ c e. C ( c .+ d ) = ( a .+ d ) ) )
13		eqeq2 2633	. . . . . . . 8 |- ( y = ( a .+ d ) -> ( ( c .+ d ) = y <-> ( c .+ d ) = ( a .+ d ) ) )
14	13	riotabidv 6613	. . . . . . 7 |- ( y = ( a .+ d ) -> ( iota_ c e. C ( c .+ d ) = y ) = ( iota_ c e. C ( c .+ d ) = ( a .+ d ) ) )
15	14	eqeq2d 2632	. . . . . 6 |- ( y = ( a .+ d ) -> ( a = ( iota_ c e. C ( c .+ d ) = y ) <-> a = ( iota_ c e. C ( c .+ d ) = ( a .+ d ) ) ) )
16	15	rspcev 3309	. . . . 5 |- ( ( ( a .+ d ) e. B /\ a = ( iota_ c e. C ( c .+ d ) = ( a .+ d ) ) ) -> E. y e. B a = ( iota_ c e. C ( c .+ d ) = y ) )
17	5, 12, 16	syl2anc 693	. . . 4 |- ( ( ( ph /\ a e. C ) /\ ( d e. D /\ ( a .+ d ) e. B ) ) -> E. y e. B a = ( iota_ c e. C ( c .+ d ) = y ) )
18		unxpwdom3.ni	. . . . . . 7 |- ( ph -> -. D ~<_ A )
19	18	adantr 481	. . . . . 6 |- ( ( ph /\ a e. C ) -> -. D ~<_ A )
20		unxpwdom3.av	. . . . . . . 8 |- ( ph -> A e. V )
21	20	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ a e. C ) /\ A. d e. D -. ( a .+ d ) e. B ) -> A e. V )
22		oveq2 6658	. . . . . . . . . . . . . 14 |- ( d = b -> ( a .+ d ) = ( a .+ b ) )
23	22	eleq1d 2686	. . . . . . . . . . . . 13 |- ( d = b -> ( ( a .+ d ) e. B <-> ( a .+ b ) e. B ) )
24	23	notbid 308	. . . . . . . . . . . 12 |- ( d = b -> ( -. ( a .+ d ) e. B <-> -. ( a .+ b ) e. B ) )
25	24	rspcv 3305	. . . . . . . . . . 11 |- ( b e. D -> ( A. d e. D -. ( a .+ d ) e. B -> -. ( a .+ b ) e. B ) )
26	25	adantl 482	. . . . . . . . . 10 |- ( ( ( ph /\ a e. C ) /\ b e. D ) -> ( A. d e. D -. ( a .+ d ) e. B -> -. ( a .+ b ) e. B ) )
27		unxpwdom3.ov	. . . . . . . . . . . . . 14 |- ( ( ph /\ a e. C /\ b e. D ) -> ( a .+ b ) e. ( A u. B ) )
28	27	3expa 1265	. . . . . . . . . . . . 13 |- ( ( ( ph /\ a e. C ) /\ b e. D ) -> ( a .+ b ) e. ( A u. B ) )
29		elun 3753	. . . . . . . . . . . . 13 |- ( ( a .+ b ) e. ( A u. B ) <-> ( ( a .+ b ) e. A \/ ( a .+ b ) e. B ) )
30	28, 29	sylib 208	. . . . . . . . . . . 12 |- ( ( ( ph /\ a e. C ) /\ b e. D ) -> ( ( a .+ b ) e. A \/ ( a .+ b ) e. B ) )
31	30	orcomd 403	. . . . . . . . . . 11 |- ( ( ( ph /\ a e. C ) /\ b e. D ) -> ( ( a .+ b ) e. B \/ ( a .+ b ) e. A ) )
32	31	ord 392	. . . . . . . . . 10 |- ( ( ( ph /\ a e. C ) /\ b e. D ) -> ( -. ( a .+ b ) e. B -> ( a .+ b ) e. A ) )
33	26, 32	syld 47	. . . . . . . . 9 |- ( ( ( ph /\ a e. C ) /\ b e. D ) -> ( A. d e. D -. ( a .+ d ) e. B -> ( a .+ b ) e. A ) )
34	33	impancom 456	. . . . . . . 8 |- ( ( ( ph /\ a e. C ) /\ A. d e. D -. ( a .+ d ) e. B ) -> ( b e. D -> ( a .+ b ) e. A ) )
35		unxpwdom3.lc	. . . . . . . . . 10 |- ( ( ( ph /\ a e. C ) /\ ( b e. D /\ c e. D ) ) -> ( ( a .+ b ) = ( a .+ c ) <-> b = c ) )
36	35	ex 450	. . . . . . . . 9 |- ( ( ph /\ a e. C ) -> ( ( b e. D /\ c e. D ) -> ( ( a .+ b ) = ( a .+ c ) <-> b = c ) ) )
37	36	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ a e. C ) /\ A. d e. D -. ( a .+ d ) e. B ) -> ( ( b e. D /\ c e. D ) -> ( ( a .+ b ) = ( a .+ c ) <-> b = c ) ) )
38	34, 37	dom2d 7996	. . . . . . 7 |- ( ( ( ph /\ a e. C ) /\ A. d e. D -. ( a .+ d ) e. B ) -> ( A e. V -> D ~<_ A ) )
39	21, 38	mpd 15	. . . . . 6 |- ( ( ( ph /\ a e. C ) /\ A. d e. D -. ( a .+ d ) e. B ) -> D ~<_ A )
40	19, 39	mtand 691	. . . . 5 |- ( ( ph /\ a e. C ) -> -. A. d e. D -. ( a .+ d ) e. B )
41		dfrex2 2996	. . . . 5 |- ( E. d e. D ( a .+ d ) e. B <-> -. A. d e. D -. ( a .+ d ) e. B )
42	40, 41	sylibr 224	. . . 4 |- ( ( ph /\ a e. C ) -> E. d e. D ( a .+ d ) e. B )
43	17, 42	reximddv 3018	. . 3 |- ( ( ph /\ a e. C ) -> E. d e. D E. y e. B a = ( iota_ c e. C ( c .+ d ) = y ) )
44		vex 3203	. . . . . . . . 9 |- d e. _V
45		vex 3203	. . . . . . . . 9 |- y e. _V
46	44, 45	op1std 7178	. . . . . . . 8 |- ( x = <. d , y >. -> ( 1st ` x ) = d )
47	46	oveq2d 6666	. . . . . . 7 |- ( x = <. d , y >. -> ( c .+ ( 1st ` x ) ) = ( c .+ d ) )
48	44, 45	op2ndd 7179	. . . . . . 7 |- ( x = <. d , y >. -> ( 2nd ` x ) = y )
49	47, 48	eqeq12d 2637	. . . . . 6 |- ( x = <. d , y >. -> ( ( c .+ ( 1st ` x ) ) = ( 2nd ` x ) <-> ( c .+ d ) = y ) )
50	49	riotabidv 6613	. . . . 5 |- ( x = <. d , y >. -> ( iota_ c e. C ( c .+ ( 1st ` x ) ) = ( 2nd ` x ) ) = ( iota_ c e. C ( c .+ d ) = y ) )
51	50	eqeq2d 2632	. . . 4 |- ( x = <. d , y >. -> ( a = ( iota_ c e. C ( c .+ ( 1st ` x ) ) = ( 2nd ` x ) ) <-> a = ( iota_ c e. C ( c .+ d ) = y ) ) )
52	51	rexxp 5264	. . 3 |- ( E. x e. ( D X. B ) a = ( iota_ c e. C ( c .+ ( 1st ` x ) ) = ( 2nd ` x ) ) <-> E. d e. D E. y e. B a = ( iota_ c e. C ( c .+ d ) = y ) )
53	43, 52	sylibr 224	. 2 |- ( ( ph /\ a e. C ) -> E. x e. ( D X. B ) a = ( iota_ c e. C ( c .+ ( 1st ` x ) ) = ( 2nd ` x ) ) )
54	4, 53	wdomd 8486	1 |- ( ph -> C ~<_* ( D X. B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   u. cun 3572   <.cop 4183   class class class wbr 4653   X. cxp 5112   ` cfv 5888   iota_crio 6610  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   ~<_ cdom 7953   ~<_^* cwdom 8462

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-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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-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-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-wdom 8464

This theorem is referenced by:  isnumbasgrplem2  37674