Theorem ltexprlemrl 7829

Description: Lemma for ltexpri 7832. 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 7724	. . . . . . . 8 |- <P C_ ( P. X. P. )
2	1	brel 4778	. . . . . . 7 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
3	2	simprd 114	. . . . . 6 |- ( A <P B -> B e. P. )
4		prop 7694	. . . . . 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 7709	. . . . 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 7694	. . . . . . . 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 7722	. . . . . . 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 535	. . . . . . . . . . 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 537	. . . . . . . . . 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 7700	. . . . . . . . . . 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 7700	. . . . . . . . . . 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 7615	. . . . . . . . 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 7627	. . . . . . . . . . . . 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 533	. . . . . . . . . . . 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 531	. . . . . . . . . . . . . 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 538	. . . . . . . . . . . . . . . . . 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 7704	. . . . . . . . . . . . . . . . . . 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 537	. . . . . . . . . . . . . . . . . 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 7600	. . . . . . . . . . . . . . . . . 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 7601	. . . . . . . . . . . . . . . . . 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 6202	. . . . . . . . . . . . . . . 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 538	. . . . . . . . . . . . . . . . . 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 6024	. . . . . . . . . . . . . . . . . . 19 |- ( ( z +Q s ) = w -> ( ( z +Q s ) +Q v ) = ( w +Q v ) )
49	48	eleq1d 2300	. . . . . . . . . . . . . . . . . 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 2308	. . . . . . . . . . . . . . 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 2294	. . . . . . . . . . . . . . . . . 18 |- ( y = ( z +Q v ) -> ( y e. ( 2nd ` A ) <-> ( z +Q v ) e. ( 2nd ` A ) ) )
54		oveq1 6024	. . . . . . . . . . . . . . . . . . 19 |- ( y = ( z +Q v ) -> ( y +Q s ) = ( ( z +Q v ) +Q s ) )
55	54	eleq1d 2300	. . . . . . . . . . . . . . . . . 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 2894	. . . . . . . . . . . . . . . 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 581	. . . . . . . . . . . . . . 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 7817	. . . . . . . . . . . . . 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 7827	. . . . . . . . . . . . . . . 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 7687	. . . . . . . . . . . . . . 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 7594	. . . . . . . . . . . . . . 15 |- ( ( f e. Q. /\ v e. Q. ) -> ( f +Q v ) e. Q. )
70	68, 69	genpprecll 7733	. . . . . . . . . . . . . 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 2309	. . . . . . . . . . 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 2655	. . . . . . . . . 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 2309	. . . . . . . . . . 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 7816	. . . . . . . . . . . . 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 7808	. . . . . . . . . . . . 13 |- ( A <P ( A +P. C ) -> ( 1st ` A ) C_ ( 1st ` ( A +P. C ) ) )
83	82	sseld 3226	. . . . . . . . . . . 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 7703	. . . . . . . . . . . 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 1340	. . . . . . . 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 2658	. . . . 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 2655	. . 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 3233	1 |- ( A <P B -> ( 1st ` B ) C_ ( 1st ` ( A +P. C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 1003   /\ w3a 1004   = wceq 1397   E.wex 1540   e. wcel 2202   E.wrex 2511   {crab 2514   C_ wss 3200   <.cop 3672   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   1stc1st 6300   2ndc2nd 6301   Q.cnq 7499   +Q cplq 7501   <Q cltq 7504   P.cnp 7510   +P. cpp 7512   <P cltp 7514

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-1o 6581  df-2o 6582  df-oadd 6585  df-omul 6586  df-er 6701  df-ec 6703  df-qs 6707  df-ni 7523  df-pli 7524  df-mi 7525  df-lti 7526  df-plpq 7563  df-mpq 7564  df-enq 7566  df-nqqs 7567  df-plqqs 7568  df-mqqs 7569  df-1nqqs 7570  df-rq 7571  df-ltnqqs 7572  df-enq0 7643  df-nq0 7644  df-0nq0 7645  df-plq0 7646  df-mq0 7647  df-inp 7685  df-iplp 7687  df-iltp 7689

This theorem is referenced by:  ltexpri  7832