Theorem ltexprlemrl 7442

Description: Lemma for ltexpri 7445. 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 7337	. . . . . . . 8 |- <P C_ ( P. X. P. )
2	1	brel 4599	. . . . . . 7 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
3	2	simprd 113	. . . . . 6 |- ( A <P B -> B e. P. )
4		prop 7307	. . . . . 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 7322	. . . . 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 281	. . . 4 |- ( ( A <P B /\ w e. ( 1st ` B ) ) -> E. v e. Q. ( w +Q v ) e. ( 1st ` B ) )
8	2	simpld 111	. . . . . . . 8 |- ( A <P B -> A e. P. )
9		prop 7307	. . . . . . . 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 7335	. . . . . . 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 281	. . . . . 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 501	. . . . 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 523	. . . . . . . . . . 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 274	. . . . . . . . . 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 525	. . . . . . . . . 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 7313	. . . . . . . . . . 11 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
18	10, 17	sylan 281	. . . . . . . . . 10 |- ( ( A <P B /\ z e. ( 1st ` A ) ) -> z e. Q. )
19	15, 16, 18	syl2anc 409	. . . . . . . . 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 7313	. . . . . . . . . . 11 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ w e. ( 1st ` B ) ) -> w e. Q. )
21	5, 20	sylan 281	. . . . . . . . . 10 |- ( ( A <P B /\ w e. ( 1st ` B ) ) -> w e. Q. )
22	21	ad3antrrr 484	. . . . . . . . 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 7228	. . . . . . . . 9 |- ( ( z e. Q. /\ w e. Q. ) -> ( z <Q w \/ z = w \/ w <Q z ) )
24	19, 22, 23	syl2anc 409	. . . . . . . 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 7240	. . . . . . . . . . . . 13 |- ( ( z e. Q. /\ w e. Q. ) -> ( z <Q w <-> E. s e. Q. ( z +Q s ) = w ) )
26	19, 22, 25	syl2anc 409	. . . . . . . . . . . 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 294	. . . . . . . . . . 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 522	. . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 521	. . . . . . . . . . . . . 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 109	. . . . . . . . . . . . . . . . 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 526	. . . . . . . . . . . . . . . . . 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 7317	. . . . . . . . . . . . . . . . . . 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 281	. . . . . . . . . . . . . . . . . 18 |- ( ( A <P B /\ u e. ( 2nd ` A ) ) -> ( u <Q ( z +Q v ) -> ( z +Q v ) e. ( 2nd ` A ) ) )
35	15, 32, 34	syl2anc 409	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 525	. . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . 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 7213	. . . . . . . . . . . . . . . . . 18 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
42	41	adantl 275	. . . . . . . . . . . . . . . . 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 7214	. . . . . . . . . . . . . . . . . 18 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( ( f +Q g ) +Q h ) = ( f +Q ( g +Q h ) ) )
44	43	adantl 275	. . . . . . . . . . . . . . . . 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 5959	. . . . . . . . . . . . . . . 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 526	. . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . 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 5789	. . . . . . . . . . . . . . . . . . 19 |- ( ( z +Q s ) = w -> ( ( z +Q s ) +Q v ) = ( w +Q v ) )
49	48	eleq1d 2209	. . . . . . . . . . . . . . . . . 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 166	. . . . . . . . . . . . . . . 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 2217	. . . . . . . . . . . . . . 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 2203	. . . . . . . . . . . . . . . . . 18 |- ( y = ( z +Q v ) -> ( y e. ( 2nd ` A ) <-> ( z +Q v ) e. ( 2nd ` A ) ) )
54		oveq1 5789	. . . . . . . . . . . . . . . . . . 19 |- ( y = ( z +Q v ) -> ( y +Q s ) = ( ( z +Q v ) +Q s ) )
55	54	eleq1d 2209	. . . . . . . . . . . . . . . . . 18 |- ( y = ( z +Q v ) -> ( ( y +Q s ) e. ( 1st ` B ) <-> ( ( z +Q v ) +Q s ) e. ( 1st ` B ) ) )
56	53, 55	anbi12d 465	. . . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . . . 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 569	. . . . . . . . . . . . . . 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 409	. . . . . . . . . . . . . 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 7430	. . . . . . . . . . . . . 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 414	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 7440	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 7300	. . . . . . . . . . . . . . 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 7207	. . . . . . . . . . . . . . 15 |- ( ( f e. Q. /\ v e. Q. ) -> ( f +Q v ) e. Q. )
70	68, 69	genpprecll 7346	. . . . . . . . . . . . . 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 409	. . . . . . . . . . . . 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 430	. . . . . . . . . . . 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 2218	. . . . . . . . . . 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 2557	. . . . . . . . . 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 114	. . . . . . . . 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 480	. . . . . . . . . . 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 109	. . . . . . . . . . . 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 274	. . . . . . . . . . . 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 2218	. . . . . . . . . . 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 7429	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ C e. P. ) -> A <P ( A +P. C ) )
81	8, 65, 80	syl2anc 409	. . . . . . . . . . . 12 |- ( A <P B -> A <P ( A +P. C ) )
82		ltprordil 7421	. . . . . . . . . . . . 13 |- ( A <P ( A +P. C ) -> ( 1st ` A ) C_ ( 1st ` ( A +P. C ) ) )
83	82	sseld 3101	. . . . . . . . . . . 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 114	. . . . . . . . 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 7316	. . . . . . . . . . . 12 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) ) -> ( w <Q z -> w e. ( 1st ` A ) ) )
88	10, 87	sylan 281	. . . . . . . . . . 11 |- ( ( A <P B /\ z e. ( 1st ` A ) ) -> ( w <Q z -> w e. ( 1st ` A ) ) )
89	15, 16, 88	syl2anc 409	. . . . . . . . . 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 1283	. . . . . . . 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 114	. . . . . 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 2560	. . . . 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 2557	. . 3 |- ( ( A <P B /\ w e. ( 1st ` B ) ) -> w e. ( 1st ` ( A +P. C ) ) )
97	96	ex 114	. 2 |- ( A <P B -> ( w e. ( 1st ` B ) -> w e. ( 1st ` ( A +P. C ) ) ) )
98	97	ssrdv 3108	1 |- ( A <P B -> ( 1st ` B ) C_ ( 1st ` ( A +P. C ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ w3o 962   /\ w3a 963   = wceq 1332   E.wex 1469   e. wcel 1481   E.wrex 2418   {crab 2421   C_ wss 3076   <.cop 3535   class class class wbr 3937   ` cfv 5131  (class class class)co 5782   1stc1st 6044   2ndc2nd 6045   Q.cnq 7112   +Q cplq 7114   <Q cltq 7117   P.cnp 7123   +P. cpp 7125   <P cltp 7127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-2o 6322  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185  df-enq0 7256  df-nq0 7257  df-0nq0 7258  df-plq0 7259  df-mq0 7260  df-inp 7298  df-iplp 7300  df-iltp 7302

This theorem is referenced by:  ltexpri  7445