Theorem ltexprlemrl 7608

Description: Lemma for ltexpri 7611. Reverse direction of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)

Hypothesis

Ref	Expression
ltexprlem.1	|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.

Assertion

Ref	Expression
ltexprlemrl	|- ( A <P B -> ( 1st ` B ) C_ ( 1st ` ( A +P. C ) ) )

Distinct variable groups:   x, y, A   x, B, y   x, C, y

Proof of Theorem ltexprlemrl

Dummy variables z w u v f g h s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelpr 7503	. . . . . . . 8 |- <P C_ ( P. X. P. )
2	1	brel 4678	. . . . . . 7 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
3	2	simprd 114	. . . . . 6 |- ( A <P B -> B e. P. )
4		prop 7473	. . . . . 6 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
5	3, 4	syl 14	. . . . 5 |- ( A <P B -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
6		prnmaddl 7488	. . . . 5 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ w e. ( 1st ` B ) ) -> E. v e. Q. ( w +Q v ) e. ( 1st ` B ) )
7	5, 6	sylan 283	. . . 4 |- ( ( A <P B /\ w e. ( 1st ` B ) ) -> E. v e. Q. ( w +Q v ) e. ( 1st ` B ) )
8	2	simpld 112	. . . . . . . 8 |- ( A <P B -> A e. P. )
9		prop 7473	. . . . . . . 8 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
10	8, 9	syl 14	. . . . . . 7 |- ( A <P B -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
11		prarloc 7501	. . . . . . 7 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ v e. Q. ) -> E. z e. ( 1st ` A ) E. u e. ( 2nd ` A ) u <Q ( z +Q v ) )
12	10, 11	sylan 283	. . . . . 6 |- ( ( A <P B /\ v e. Q. ) -> E. z e. ( 1st ` A ) E. u e. ( 2nd ` A ) u <Q ( z +Q v ) )
13	12	ad2ant2r 509	. . . . 5 |- ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) -> E. z e. ( 1st ` A ) E. u e. ( 2nd ` A ) u <Q ( z +Q v ) )
14		simplll 533	. . . . . . . . . . 11 |- ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) -> A <P B )
15	14	adantr 276	. . . . . . . . . 10 |- ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> A <P B )
16		simplrl 535	. . . . . . . . . 10 |- ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> z e. ( 1st ` A ) )
17		elprnql 7479	. . . . . . . . . . 11 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
18	10, 17	sylan 283	. . . . . . . . . 10 |- ( ( A <P B /\ z e. ( 1st ` A ) ) -> z e. Q. )
19	15, 16, 18	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> z e. Q. )
20		elprnql 7479	. . . . . . . . . . 11 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ w e. ( 1st ` B ) ) -> w e. Q. )
21	5, 20	sylan 283	. . . . . . . . . 10 |- ( ( A <P B /\ w e. ( 1st ` B ) ) -> w e. Q. )
22	21	ad3antrrr 492	. . . . . . . . 9 |- ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> w e. Q. )
23		nqtri3or 7394	. . . . . . . . 9 |- ( ( z e. Q. /\ w e. Q. ) -> ( z <Q w \/ z = w \/ w <Q z ) )
24	19, 22, 23	syl2anc 411	. . . . . . . 8 |- ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> ( z <Q w \/ z = w \/ w <Q z ) )
25		ltexnqq 7406	. . . . . . . . . . . . 13 |- ( ( z e. Q. /\ w e. Q. ) -> ( z <Q w <-> E. s e. Q. ( z +Q s ) = w ) )
26	19, 22, 25	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> ( z <Q w <-> E. s e. Q. ( z +Q s ) = w ) )
27	26	biimpa 296	. . . . . . . . . . 11 |- ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z <Q w ) -> E. s e. Q. ( z +Q s ) = w )
28		simprr 531	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z <Q w ) /\ ( s e. Q. /\ ( z +Q s ) = w ) ) -> ( z +Q s ) = w )
29	16	ad2antrr 488	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z <Q w ) /\ ( s e. Q. /\ ( z +Q s ) = w ) ) -> z e. ( 1st ` A ) )
30		simprl 529	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z <Q w ) /\ ( s e. Q. /\ ( z +Q s ) = w ) ) -> s e. Q. )
31		simpr 110	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> u <Q ( z +Q v ) )
32		simplrr 536	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> u e. ( 2nd ` A ) )
33		prcunqu 7483	. . . . . . . . . . . . . . . . . . 19 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ u e. ( 2nd ` A ) ) -> ( u <Q ( z +Q v ) -> ( z +Q v ) e. ( 2nd ` A ) ) )
34	10, 33	sylan 283	. . . . . . . . . . . . . . . . . 18 |- ( ( A <P B /\ u e. ( 2nd ` A ) ) -> ( u <Q ( z +Q v ) -> ( z +Q v ) e. ( 2nd ` A ) ) )
35	15, 32, 34	syl2anc 411	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> ( u <Q ( z +Q v ) -> ( z +Q v ) e. ( 2nd ` A ) ) )
36	31, 35	mpd 13	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> ( z +Q v ) e. ( 2nd ` A ) )
37	36	ad2antrr 488	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z <Q w ) /\ ( s e. Q. /\ ( z +Q s ) = w ) ) -> ( z +Q v ) e. ( 2nd ` A ) )
38	19	ad2antrr 488	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z <Q w ) /\ ( s e. Q. /\ ( z +Q s ) = w ) ) -> z e. Q. )
39		simplrl 535	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) -> v e. Q. )
40	39	ad3antrrr 492	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z <Q w ) /\ ( s e. Q. /\ ( z +Q s ) = w ) ) -> v e. Q. )
41		addcomnqg 7379	. . . . . . . . . . . . . . . . . 18 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
42	41	adantl 277	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z <Q w ) /\ ( s e. Q. /\ ( z +Q s ) = w ) ) /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
43		addassnqg 7380	. . . . . . . . . . . . . . . . . 18 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( ( f +Q g ) +Q h ) = ( f +Q ( g +Q h ) ) )
44	43	adantl 277	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z <Q w ) /\ ( s e. Q. /\ ( z +Q s ) = w ) ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( ( f +Q g ) +Q h ) = ( f +Q ( g +Q h ) ) )
45	38, 40, 30, 42, 44	caov32d 6054	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z <Q w ) /\ ( s e. Q. /\ ( z +Q s ) = w ) ) -> ( ( z +Q v ) +Q s ) = ( ( z +Q s ) +Q v ) )
46		simplrr 536	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) -> ( w +Q v ) e. ( 1st ` B ) )
47	46	ad3antrrr 492	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z <Q w ) /\ ( s e. Q. /\ ( z +Q s ) = w ) ) -> ( w +Q v ) e. ( 1st ` B ) )
48		oveq1 5881	. . . . . . . . . . . . . . . . . . 19 |- ( ( z +Q s ) = w -> ( ( z +Q s ) +Q v ) = ( w +Q v ) )
49	48	eleq1d 2246	. . . . . . . . . . . . . . . . . 18 |- ( ( z +Q s ) = w -> ( ( ( z +Q s ) +Q v ) e. ( 1st ` B ) <-> ( w +Q v ) e. ( 1st ` B ) ) )
50	28, 49	syl 14	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z <Q w ) /\ ( s e. Q. /\ ( z +Q s ) = w ) ) -> ( ( ( z +Q s ) +Q v ) e. ( 1st ` B ) <-> ( w +Q v ) e. ( 1st ` B ) ) )
51	47, 50	mpbird 167	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z <Q w ) /\ ( s e. Q. /\ ( z +Q s ) = w ) ) -> ( ( z +Q s ) +Q v ) e. ( 1st ` B ) )
52	45, 51	eqeltrd 2254	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z <Q w ) /\ ( s e. Q. /\ ( z +Q s ) = w ) ) -> ( ( z +Q v ) +Q s ) e. ( 1st ` B ) )
53		eleq1 2240	. . . . . . . . . . . . . . . . . 18 |- ( y = ( z +Q v ) -> ( y e. ( 2nd ` A ) <-> ( z +Q v ) e. ( 2nd ` A ) ) )
54		oveq1 5881	. . . . . . . . . . . . . . . . . . 19 |- ( y = ( z +Q v ) -> ( y +Q s ) = ( ( z +Q v ) +Q s ) )
55	54	eleq1d 2246	. . . . . . . . . . . . . . . . . 18 |- ( y = ( z +Q v ) -> ( ( y +Q s ) e. ( 1st ` B ) <-> ( ( z +Q v ) +Q s ) e. ( 1st ` B ) ) )
56	53, 55	anbi12d 473	. . . . . . . . . . . . . . . . 17 |- ( y = ( z +Q v ) -> ( ( y e. ( 2nd ` A ) /\ ( y +Q s ) e. ( 1st ` B ) ) <-> ( ( z +Q v ) e. ( 2nd ` A ) /\ ( ( z +Q v ) +Q s ) e. ( 1st ` B ) ) ) )
57	56	spcegv 2825	. . . . . . . . . . . . . . . 16 |- ( ( z +Q v ) e. ( 2nd ` A ) -> ( ( ( z +Q v ) e. ( 2nd ` A ) /\ ( ( z +Q v ) +Q s ) e. ( 1st ` B ) ) -> E. y ( y e. ( 2nd ` A ) /\ ( y +Q s ) e. ( 1st ` B ) ) ) )
58	57	anabsi5 579	. . . . . . . . . . . . . . 15 |- ( ( ( z +Q v ) e. ( 2nd ` A ) /\ ( ( z +Q v ) +Q s ) e. ( 1st ` B ) ) -> E. y ( y e. ( 2nd ` A ) /\ ( y +Q s ) e. ( 1st ` B ) ) )
59	37, 52, 58	syl2anc 411	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z <Q w ) /\ ( s e. Q. /\ ( z +Q s ) = w ) ) -> E. y ( y e. ( 2nd ` A ) /\ ( y +Q s ) e. ( 1st ` B ) ) )
60		ltexprlem.1	. . . . . . . . . . . . . . 15 |- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.
61	60	ltexprlemell 7596	. . . . . . . . . . . . . 14 |- ( s e. ( 1st ` C ) <-> ( s e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q s ) e. ( 1st ` B ) ) ) )
62	30, 59, 61	sylanbrc 417	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z <Q w ) /\ ( s e. Q. /\ ( z +Q s ) = w ) ) -> s e. ( 1st ` C ) )
63	15, 8	syl 14	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> A e. P. )
64	63	ad2antrr 488	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z <Q w ) /\ ( s e. Q. /\ ( z +Q s ) = w ) ) -> A e. P. )
65	60	ltexprlempr 7606	. . . . . . . . . . . . . . . 16 |- ( A <P B -> C e. P. )
66	15, 65	syl 14	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> C e. P. )
67	66	ad2antrr 488	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z <Q w ) /\ ( s e. Q. /\ ( z +Q s ) = w ) ) -> C e. P. )
68		df-iplp 7466	. . . . . . . . . . . . . . 15 |- +P. = ( x e. P. , w e. P. |-> <. { z e. Q. | E. f e. Q. E. v e. Q. ( f e. ( 1st ` x ) /\ v e. ( 1st ` w ) /\ z = ( f +Q v ) ) } , { z e. Q. | E. f e. Q. E. v e. Q. ( f e. ( 2nd ` x ) /\ v e. ( 2nd ` w ) /\ z = ( f +Q v ) ) } >. )
69		addclnq 7373	. . . . . . . . . . . . . . 15 |- ( ( f e. Q. /\ v e. Q. ) -> ( f +Q v ) e. Q. )
70	68, 69	genpprecll 7512	. . . . . . . . . . . . . 14 |- ( ( A e. P. /\ C e. P. ) -> ( ( z e. ( 1st ` A ) /\ s e. ( 1st ` C ) ) -> ( z +Q s ) e. ( 1st ` ( A +P. C ) ) ) )
71	64, 67, 70	syl2anc 411	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z <Q w ) /\ ( s e. Q. /\ ( z +Q s ) = w ) ) -> ( ( z e. ( 1st ` A ) /\ s e. ( 1st ` C ) ) -> ( z +Q s ) e. ( 1st ` ( A +P. C ) ) ) )
72	29, 62, 71	mp2and 433	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z <Q w ) /\ ( s e. Q. /\ ( z +Q s ) = w ) ) -> ( z +Q s ) e. ( 1st ` ( A +P. C ) ) )
73	28, 72	eqeltrrd 2255	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z <Q w ) /\ ( s e. Q. /\ ( z +Q s ) = w ) ) -> w e. ( 1st ` ( A +P. C ) ) )
74	27, 73	rexlimddv 2599	. . . . . . . . . 10 |- ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z <Q w ) -> w e. ( 1st ` ( A +P. C ) ) )
75	74	ex 115	. . . . . . . . 9 |- ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> ( z <Q w -> w e. ( 1st ` ( A +P. C ) ) ) )
76	14	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z = w ) -> A <P B )
77		simpr 110	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z = w ) -> z = w )
78	16	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z = w ) -> z e. ( 1st ` A ) )
79	77, 78	eqeltrrd 2255	. . . . . . . . . . 11 |- ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z = w ) -> w e. ( 1st ` A ) )
80		ltaddpr 7595	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ C e. P. ) -> A <P ( A +P. C ) )
81	8, 65, 80	syl2anc 411	. . . . . . . . . . . 12 |- ( A <P B -> A <P ( A +P. C ) )
82		ltprordil 7587	. . . . . . . . . . . . 13 |- ( A <P ( A +P. C ) -> ( 1st ` A ) C_ ( 1st ` ( A +P. C ) ) )
83	82	sseld 3154	. . . . . . . . . . . 12 |- ( A <P ( A +P. C ) -> ( w e. ( 1st ` A ) -> w e. ( 1st ` ( A +P. C ) ) ) )
84	81, 83	syl 14	. . . . . . . . . . 11 |- ( A <P B -> ( w e. ( 1st ` A ) -> w e. ( 1st ` ( A +P. C ) ) ) )
85	76, 79, 84	sylc 62	. . . . . . . . . 10 |- ( ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ z = w ) -> w e. ( 1st ` ( A +P. C ) ) )
86	85	ex 115	. . . . . . . . 9 |- ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> ( z = w -> w e. ( 1st ` ( A +P. C ) ) ) )
87		prcdnql 7482	. . . . . . . . . . . 12 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) ) -> ( w <Q z -> w e. ( 1st ` A ) ) )
88	10, 87	sylan 283	. . . . . . . . . . 11 |- ( ( A <P B /\ z e. ( 1st ` A ) ) -> ( w <Q z -> w e. ( 1st ` A ) ) )
89	15, 16, 88	syl2anc 411	. . . . . . . . . 10 |- ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> ( w <Q z -> w e. ( 1st ` A ) ) )
90	15, 89, 84	sylsyld 58	. . . . . . . . 9 |- ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> ( w <Q z -> w e. ( 1st ` ( A +P. C ) ) ) )
91	75, 86, 90	3jaod 1304	. . . . . . . 8 |- ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> ( ( z <Q w \/ z = w \/ w <Q z ) -> w e. ( 1st ` ( A +P. C ) ) ) )
92	24, 91	mpd 13	. . . . . . 7 |- ( ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> w e. ( 1st ` ( A +P. C ) ) )
93	92	ex 115	. . . . . 6 |- ( ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) -> ( u <Q ( z +Q v ) -> w e. ( 1st ` ( A +P. C ) ) ) )
94	93	rexlimdvva 2602	. . . . 5 |- ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) -> ( E. z e. ( 1st ` A ) E. u e. ( 2nd ` A ) u <Q ( z +Q v ) -> w e. ( 1st ` ( A +P. C ) ) ) )
95	13, 94	mpd 13	. . . 4 |- ( ( ( A <P B /\ w e. ( 1st ` B ) ) /\ ( v e. Q. /\ ( w +Q v ) e. ( 1st ` B ) ) ) -> w e. ( 1st ` ( A +P. C ) ) )
96	7, 95	rexlimddv 2599	. . 3 |- ( ( A <P B /\ w e. ( 1st ` B ) ) -> w e. ( 1st ` ( A +P. C ) ) )
97	96	ex 115	. 2 |- ( A <P B -> ( w e. ( 1st ` B ) -> w e. ( 1st ` ( A +P. C ) ) ) )
98	97	ssrdv 3161	1 |- ( A <P B -> ( 1st ` B ) C_ ( 1st ` ( A +P. C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 977   /\ w3a 978   = wceq 1353   E.wex 1492   e. wcel 2148   E.wrex 2456   {crab 2459   C_ wss 3129   <.cop 3595   class class class wbr 4003   ` cfv 5216  (class class class)co 5874   1stc1st 6138   2ndc2nd 6139   Q.cnq 7278   +Q cplq 7280   <Q cltq 7283   P.cnp 7289   +P. cpp 7291   <P cltp 7293

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-2o 6417  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-enq0 7422  df-nq0 7423  df-0nq0 7424  df-plq0 7425  df-mq0 7426  df-inp 7464  df-iplp 7466  df-iltp 7468

This theorem is referenced by:  ltexpri  7611