Theorem ltexprlemru 7927

Description: Lemma for ltexpri 7928. One direction of our result for upper 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
ltexprlemru	|- ( A <P B -> ( 2nd ` B ) C_ ( 2nd ` ( A +P. C ) ) )

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

Proof of Theorem ltexprlemru

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

Step	Hyp	Ref	Expression
1		ltrelpr 7820	. . . . . . . 8 |- <P C_ ( P. X. P. )
2	1	brel 4802	. . . . . . 7 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
3	2	simprd 114	. . . . . 6 |- ( A <P B -> B e. P. )
4		prop 7790	. . . . . 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		prnminu 7804	. . . . 5 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ w e. ( 2nd ` B ) ) -> E. t e. ( 2nd ` B ) t <Q w )
7	5, 6	sylan 283	. . . 4 |- ( ( A <P B /\ w e. ( 2nd ` B ) ) -> E. t e. ( 2nd ` B ) t <Q w )
8		simprr 533	. . . . . 6 |- ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) -> t <Q w )
9		elprnqu 7797	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ t e. ( 2nd ` B ) ) -> t e. Q. )
10	5, 9	sylan 283	. . . . . . . 8 |- ( ( A <P B /\ t e. ( 2nd ` B ) ) -> t e. Q. )
11	10	ad2ant2r 509	. . . . . . 7 |- ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) -> t e. Q. )
12		elprnqu 7797	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ w e. ( 2nd ` B ) ) -> w e. Q. )
13	5, 12	sylan 283	. . . . . . . 8 |- ( ( A <P B /\ w e. ( 2nd ` B ) ) -> w e. Q. )
14	13	adantr 276	. . . . . . 7 |- ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) -> w e. Q. )
15		ltexnqq 7723	. . . . . . 7 |- ( ( t e. Q. /\ w e. Q. ) -> ( t <Q w <-> E. v e. Q. ( t +Q v ) = w ) )
16	11, 14, 15	syl2anc 411	. . . . . 6 |- ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) -> ( t <Q w <-> E. v e. Q. ( t +Q v ) = w ) )
17	8, 16	mpbid 147	. . . . 5 |- ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) -> E. v e. Q. ( t +Q v ) = w )
18	2	simpld 112	. . . . . . . . . 10 |- ( A <P B -> A e. P. )
19		prop 7790	. . . . . . . . . 10 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
20	18, 19	syl 14	. . . . . . . . 9 |- ( A <P B -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
21		prarloc 7818	. . . . . . . . 9 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ v e. Q. ) -> E. z e. ( 1st ` A ) E. u e. ( 2nd ` A ) u <Q ( z +Q v ) )
22	20, 21	sylan 283	. . . . . . . 8 |- ( ( A <P B /\ v e. Q. ) -> E. z e. ( 1st ` A ) E. u e. ( 2nd ` A ) u <Q ( z +Q v ) )
23	22	adantlr 477	. . . . . . 7 |- ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ v e. Q. ) -> E. z e. ( 1st ` A ) E. u e. ( 2nd ` A ) u <Q ( z +Q v ) )
24	23	ad2ant2r 509	. . . . . 6 |- ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) -> E. z e. ( 1st ` A ) E. u e. ( 2nd ` A ) u <Q ( z +Q v ) )
25		simplll 535	. . . . . . . . . . . . 13 |- ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) -> A <P B )
26	25	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> A <P B )
27		ltdfpr 7821	. . . . . . . . . . . . . 14 |- ( ( A e. P. /\ B e. P. ) -> ( A <P B <-> E. q e. Q. ( q e. ( 2nd ` A ) /\ q e. ( 1st ` B ) ) ) )
28	27	biimpd 144	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ B e. P. ) -> ( A <P B -> E. q e. Q. ( q e. ( 2nd ` A ) /\ q e. ( 1st ` B ) ) ) )
29	2, 28	mpcom 36	. . . . . . . . . . . 12 |- ( A <P B -> E. q e. Q. ( q e. ( 2nd ` A ) /\ q e. ( 1st ` B ) ) )
30	26, 29	syl 14	. . . . . . . . . . 11 |- ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> E. q e. Q. ( q e. ( 2nd ` A ) /\ q e. ( 1st ` B ) ) )
31	25	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) -> A <P B )
32	31	ad2antrr 488	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( q e. Q. /\ ( q e. ( 2nd ` A ) /\ q e. ( 1st ` B ) ) ) ) -> A <P B )
33		simplrl 537	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> z e. ( 1st ` A ) )
34	33	adantr 276	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( q e. Q. /\ ( q e. ( 2nd ` A ) /\ q e. ( 1st ` B ) ) ) ) -> z e. ( 1st ` A ) )
35		simprrl 541	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( q e. Q. /\ ( q e. ( 2nd ` A ) /\ q e. ( 1st ` B ) ) ) ) -> q e. ( 2nd ` A ) )
36		prltlu 7802	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) -> z <Q q )
37	20, 36	syl3an1 1307	. . . . . . . . . . . . 13 |- ( ( A <P B /\ z e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) -> z <Q q )
38	32, 34, 35, 37	syl3anc 1274	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( q e. Q. /\ ( q e. ( 2nd ` A ) /\ q e. ( 1st ` B ) ) ) ) -> z <Q q )
39		simprrr 542	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( q e. Q. /\ ( q e. ( 2nd ` A ) /\ q e. ( 1st ` B ) ) ) ) -> q e. ( 1st ` B ) )
40		simplrl 537	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) -> t e. ( 2nd ` B ) )
41	40	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) -> t e. ( 2nd ` B ) )
42	41	ad2antrr 488	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( q e. Q. /\ ( q e. ( 2nd ` A ) /\ q e. ( 1st ` B ) ) ) ) -> t e. ( 2nd ` B ) )
43		prltlu 7802	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ q e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) -> q <Q t )
44	5, 43	syl3an1 1307	. . . . . . . . . . . . 13 |- ( ( A <P B /\ q e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) -> q <Q t )
45	32, 39, 42, 44	syl3anc 1274	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( q e. Q. /\ ( q e. ( 2nd ` A ) /\ q e. ( 1st ` B ) ) ) ) -> q <Q t )
46		ltsonq 7713	. . . . . . . . . . . . 13 |- <Q Or Q.
47		ltrelnq 7680	. . . . . . . . . . . . 13 |- <Q C_ ( Q. X. Q. )
48	46, 47	sotri 5158	. . . . . . . . . . . 12 |- ( ( z <Q q /\ q <Q t ) -> z <Q t )
49	38, 45, 48	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( q e. Q. /\ ( q e. ( 2nd ` A ) /\ q e. ( 1st ` B ) ) ) ) -> z <Q t )
50	30, 49	rexlimddv 2665	. . . . . . . . . 10 |- ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> z <Q t )
51		ltexnqi 7724	. . . . . . . . . 10 |- ( z <Q t -> E. s e. Q. ( z +Q s ) = t )
52	50, 51	syl 14	. . . . . . . . 9 |- ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> E. s e. Q. ( z +Q s ) = t )
53		simplrr 538	. . . . . . . . . . . 12 |- ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) -> ( t +Q v ) = w )
54	53	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( s e. Q. /\ ( z +Q s ) = t ) ) -> ( t +Q v ) = w )
55		simprr 533	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( s e. Q. /\ ( z +Q s ) = t ) ) -> ( z +Q s ) = t )
56		oveq1 6057	. . . . . . . . . . . . 13 |- ( ( z +Q s ) = t -> ( ( z +Q s ) +Q v ) = ( t +Q v ) )
57	56	eqeq1d 2241	. . . . . . . . . . . 12 |- ( ( z +Q s ) = t -> ( ( ( z +Q s ) +Q v ) = w <-> ( t +Q v ) = w ) )
58	55, 57	syl 14	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( s e. Q. /\ ( z +Q s ) = t ) ) -> ( ( ( z +Q s ) +Q v ) = w <-> ( t +Q v ) = w ) )
59	54, 58	mpbird 167	. . . . . . . . . 10 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( s e. Q. /\ ( z +Q s ) = t ) ) -> ( ( z +Q s ) +Q v ) = w )
60		elprnql 7796	. . . . . . . . . . . . . . . . 17 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
61	20, 60	sylan 283	. . . . . . . . . . . . . . . 16 |- ( ( A <P B /\ z e. ( 1st ` A ) ) -> z e. Q. )
62	61	adantlr 477	. . . . . . . . . . . . . . 15 |- ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ z e. ( 1st ` A ) ) -> z e. Q. )
63	62	ad2ant2r 509	. . . . . . . . . . . . . 14 |- ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) -> z e. Q. )
64	63	adantlr 477	. . . . . . . . . . . . 13 |- ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) -> z e. Q. )
65	64	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( s e. Q. /\ ( z +Q s ) = t ) ) -> z e. Q. )
66		simplrl 537	. . . . . . . . . . . . 13 |- ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) -> v e. Q. )
67	66	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( s e. Q. /\ ( z +Q s ) = t ) ) -> v e. Q. )
68		simprl 531	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( s e. Q. /\ ( z +Q s ) = t ) ) -> s e. Q. )
69		addcomnqg 7696	. . . . . . . . . . . . 13 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
70	69	adantl 277	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( s e. Q. /\ ( z +Q s ) = t ) ) /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
71		addassnqg 7697	. . . . . . . . . . . . 13 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( ( f +Q g ) +Q h ) = ( f +Q ( g +Q h ) ) )
72	71	adantl 277	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( s e. Q. /\ ( z +Q s ) = t ) ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( ( f +Q g ) +Q h ) = ( f +Q ( g +Q h ) ) )
73	65, 67, 68, 70, 72	caov32d 6235	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( s e. Q. /\ ( z +Q s ) = t ) ) -> ( ( z +Q v ) +Q s ) = ( ( z +Q s ) +Q v ) )
74		simpr 110	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> u <Q ( z +Q v ) )
75		simplrr 538	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> u e. ( 2nd ` A ) )
76		prcunqu 7800	. . . . . . . . . . . . . . . 16 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ u e. ( 2nd ` A ) ) -> ( u <Q ( z +Q v ) -> ( z +Q v ) e. ( 2nd ` A ) ) )
77	20, 76	sylan 283	. . . . . . . . . . . . . . 15 |- ( ( A <P B /\ u e. ( 2nd ` A ) ) -> ( u <Q ( z +Q v ) -> ( z +Q v ) e. ( 2nd ` A ) ) )
78	26, 75, 77	syl2anc 411	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> ( u <Q ( z +Q v ) -> ( z +Q v ) e. ( 2nd ` A ) ) )
79	74, 78	mpd 13	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> ( z +Q v ) e. ( 2nd ` A ) )
80	79	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( s e. Q. /\ ( z +Q s ) = t ) ) -> ( z +Q v ) e. ( 2nd ` A ) )
81	33	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( s e. Q. /\ ( z +Q s ) = t ) ) -> z e. ( 1st ` A ) )
82	41	ad2antrr 488	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( s e. Q. /\ ( z +Q s ) = t ) ) -> t e. ( 2nd ` B ) )
83	55, 82	eqeltrd 2309	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( s e. Q. /\ ( z +Q s ) = t ) ) -> ( z +Q s ) e. ( 2nd ` B ) )
84		eleq1 2295	. . . . . . . . . . . . . . . . 17 |- ( y = z -> ( y e. ( 1st ` A ) <-> z e. ( 1st ` A ) ) )
85		oveq1 6057	. . . . . . . . . . . . . . . . . 18 |- ( y = z -> ( y +Q s ) = ( z +Q s ) )
86	85	eleq1d 2301	. . . . . . . . . . . . . . . . 17 |- ( y = z -> ( ( y +Q s ) e. ( 2nd ` B ) <-> ( z +Q s ) e. ( 2nd ` B ) ) )
87	84, 86	anbi12d 473	. . . . . . . . . . . . . . . 16 |- ( y = z -> ( ( y e. ( 1st ` A ) /\ ( y +Q s ) e. ( 2nd ` B ) ) <-> ( z e. ( 1st ` A ) /\ ( z +Q s ) e. ( 2nd ` B ) ) ) )
88	87	spcegv 2905	. . . . . . . . . . . . . . 15 |- ( z e. ( 1st ` A ) -> ( ( z e. ( 1st ` A ) /\ ( z +Q s ) e. ( 2nd ` B ) ) -> E. y ( y e. ( 1st ` A ) /\ ( y +Q s ) e. ( 2nd ` B ) ) ) )
89	88	anabsi5 581	. . . . . . . . . . . . . 14 |- ( ( z e. ( 1st ` A ) /\ ( z +Q s ) e. ( 2nd ` B ) ) -> E. y ( y e. ( 1st ` A ) /\ ( y +Q s ) e. ( 2nd ` B ) ) )
90	81, 83, 89	syl2anc 411	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( s e. Q. /\ ( z +Q s ) = t ) ) -> E. y ( y e. ( 1st ` A ) /\ ( y +Q s ) e. ( 2nd ` B ) ) )
91		ltexprlem.1	. . . . . . . . . . . . . 14 |- 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 ) ) } >.
92	91	ltexprlemelu 7914	. . . . . . . . . . . . 13 |- ( s e. ( 2nd ` C ) <-> ( s e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q s ) e. ( 2nd ` B ) ) ) )
93	68, 90, 92	sylanbrc 417	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( s e. Q. /\ ( z +Q s ) = t ) ) -> s e. ( 2nd ` C ) )
94	31	ad2antrr 488	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( s e. Q. /\ ( z +Q s ) = t ) ) -> A <P B )
95	94, 18	syl 14	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( s e. Q. /\ ( z +Q s ) = t ) ) -> A e. P. )
96	91	ltexprlempr 7923	. . . . . . . . . . . . . 14 |- ( A <P B -> C e. P. )
97	94, 96	syl 14	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( s e. Q. /\ ( z +Q s ) = t ) ) -> C e. P. )
98		df-iplp 7783	. . . . . . . . . . . . . 14 |- +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 ) ) } >. )
99		addclnq 7690	. . . . . . . . . . . . . 14 |- ( ( f e. Q. /\ v e. Q. ) -> ( f +Q v ) e. Q. )
100	98, 99	genppreclu 7830	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ C e. P. ) -> ( ( ( z +Q v ) e. ( 2nd ` A ) /\ s e. ( 2nd ` C ) ) -> ( ( z +Q v ) +Q s ) e. ( 2nd ` ( A +P. C ) ) ) )
101	95, 97, 100	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( s e. Q. /\ ( z +Q s ) = t ) ) -> ( ( ( z +Q v ) e. ( 2nd ` A ) /\ s e. ( 2nd ` C ) ) -> ( ( z +Q v ) +Q s ) e. ( 2nd ` ( A +P. C ) ) ) )
102	80, 93, 101	mp2and 433	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( s e. Q. /\ ( z +Q s ) = t ) ) -> ( ( z +Q v ) +Q s ) e. ( 2nd ` ( A +P. C ) ) )
103	73, 102	eqeltrrd 2310	. . . . . . . . . 10 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( s e. Q. /\ ( z +Q s ) = t ) ) -> ( ( z +Q s ) +Q v ) e. ( 2nd ` ( A +P. C ) ) )
104	59, 103	eqeltrrd 2310	. . . . . . . . 9 |- ( ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) /\ ( s e. Q. /\ ( z +Q s ) = t ) ) -> w e. ( 2nd ` ( A +P. C ) ) )
105	52, 104	rexlimddv 2665	. . . . . . . 8 |- ( ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) /\ u <Q ( z +Q v ) ) -> w e. ( 2nd ` ( A +P. C ) ) )
106	105	ex 115	. . . . . . 7 |- ( ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) /\ ( z e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) -> ( u <Q ( z +Q v ) -> w e. ( 2nd ` ( A +P. C ) ) ) )
107	106	rexlimdvva 2668	. . . . . 6 |- ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) -> ( E. z e. ( 1st ` A ) E. u e. ( 2nd ` A ) u <Q ( z +Q v ) -> w e. ( 2nd ` ( A +P. C ) ) ) )
108	24, 107	mpd 13	. . . . 5 |- ( ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) /\ ( v e. Q. /\ ( t +Q v ) = w ) ) -> w e. ( 2nd ` ( A +P. C ) ) )
109	17, 108	rexlimddv 2665	. . . 4 |- ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) -> w e. ( 2nd ` ( A +P. C ) ) )
110	7, 109	rexlimddv 2665	. . 3 |- ( ( A <P B /\ w e. ( 2nd ` B ) ) -> w e. ( 2nd ` ( A +P. C ) ) )
111	110	ex 115	. 2 |- ( A <P B -> ( w e. ( 2nd ` B ) -> w e. ( 2nd ` ( A +P. C ) ) ) )
112	111	ssrdv 3244	1 |- ( A <P B -> ( 2nd ` B ) C_ ( 2nd ` ( A +P. C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   E.wex 1541   e. wcel 2203   E.wrex 2521   {crab 2524   C_ wss 3211   <.cop 3692   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   1stc1st 6332   2ndc2nd 6333   Q.cnq 7595   +Q cplq 7597   <Q cltq 7600   P.cnp 7606   +P. cpp 7608   <P cltp 7610

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-2o 6648  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-mqqs 7665  df-1nqqs 7666  df-rq 7667  df-ltnqqs 7668  df-enq0 7739  df-nq0 7740  df-0nq0 7741  df-plq0 7742  df-mq0 7743  df-inp 7781  df-iplp 7783  df-iltp 7785

This theorem is referenced by:  ltexpri  7928