Theorem ltexprlemru 7979

Description: Lemma for ltexpri 7980. 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 7872	. . . . . . . 8 |- <P C_ ( P. X. P. )
2	1	brel 4827	. . . . . . 7 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
3	2	simprd 114	. . . . . 6 |- ( A <P B -> B e. P. )
4		prop 7842	. . . . . 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 7856	. . . . 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 537	. . . . . 6 |- ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) -> t <Q w )
9		elprnqu 7849	. . . . . . . . 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 513	. . . . . . 7 |- ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) -> t e. Q. )
12		elprnqu 7849	. . . . . . . . 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 7775	. . . . . . 7 |- ( ( t e. Q. /\ w e. Q. ) -> ( t <Q w <-> E. v e. Q. ( t +Q v ) = w ) )
16	11, 14, 15	syl2anc 415	. . . . . 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 7842	. . . . . . . . . 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 7870	. . . . . . . . 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 481	. . . . . . 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 513	. . . . . 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 539	. . . . . . . . . . . . 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 492	. . . . . . . . . . . 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 7873	. . . . . . . . . . . . . 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 492	. . . . . . . . . . . . 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 541	. . . . . . . . . . . . . 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 545	. . . . . . . . . . . . 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 7854	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) -> z <Q q )
37	20, 36	syl3an1 1311	. . . . . . . . . . . . 13 |- ( ( A <P B /\ z e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) -> z <Q q )
38	32, 34, 35, 37	syl3anc 1278	. . . . . . . . . . . 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 546	. . . . . . . . . . . . 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 541	. . . . . . . . . . . . . . 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 492	. . . . . . . . . . . . 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 7854	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ q e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) -> q <Q t )
44	5, 43	syl3an1 1311	. . . . . . . . . . . . 13 |- ( ( A <P B /\ q e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) -> q <Q t )
45	32, 39, 42, 44	syl3anc 1278	. . . . . . . . . . . 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 7765	. . . . . . . . . . . . 13 |- <Q Or Q.
47		ltrelnq 7732	. . . . . . . . . . . . 13 |- <Q C_ ( Q. X. Q. )
48	46, 47	sotri 5183	. . . . . . . . . . . 12 |- ( ( z <Q q /\ q <Q t ) -> z <Q t )
49	38, 45, 48	syl2anc 415	. . . . . . . . . . 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 2673	. . . . . . . . . 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 7776	. . . . . . . . . 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 542	. . . . . . . . . . . 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 492	. . . . . . . . . . 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 537	. . . . . . . . . . . 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 6092	. . . . . . . . . . . . 13 |- ( ( z +Q s ) = t -> ( ( z +Q s ) +Q v ) = ( t +Q v ) )
57	56	eqeq1d 2247	. . . . . . . . . . . 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 7848	. . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 15 |- ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ z e. ( 1st ` A ) ) -> z e. Q. )
63	62	ad2ant2r 513	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 492	. . . . . . . . . . . 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 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 ) ) ) -> v e. Q. )
67	66	ad2antrr 492	. . . . . . . . . . . 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 535	. . . . . . . . . . . 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 7748	. . . . . . . . . . . . 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 7749	. . . . . . . . . . . . 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 6270	. . . . . . . . . . 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 542	. . . . . . . . . . . . . . 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 7852	. . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . 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 492	. . . . . . . . . . . . . . 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 2315	. . . . . . . . . . . . . 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 2301	. . . . . . . . . . . . . . . . 17 |- ( y = z -> ( y e. ( 1st ` A ) <-> z e. ( 1st ` A ) ) )
85		oveq1 6092	. . . . . . . . . . . . . . . . . 18 |- ( y = z -> ( y +Q s ) = ( z +Q s ) )
86	85	eleq1d 2307	. . . . . . . . . . . . . . . . 17 |- ( y = z -> ( ( y +Q s ) e. ( 2nd ` B ) <-> ( z +Q s ) e. ( 2nd ` B ) ) )
87	84, 86	anbi12d 477	. . . . . . . . . . . . . . . 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 2913	. . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . 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 415	. . . . . . . . . . . . 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 7966	. . . . . . . . . . . . 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 421	. . . . . . . . . . . 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 492	. . . . . . . . . . . . . 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 7975	. . . . . . . . . . . . . 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 7835	. . . . . . . . . . . . . 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 7742	. . . . . . . . . . . . . 14 |- ( ( f e. Q. /\ v e. Q. ) -> ( f +Q v ) e. Q. )
100	98, 99	genppreclu 7882	. . . . . . . . . . . . 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 415	. . . . . . . . . . . 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 437	. . . . . . . . . . 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 2316	. . . . . . . . . 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 2316	. . . . . . . . 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 2673	. . . . . . . 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 2676	. . . . . 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 2673	. . . 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 2673	. . 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 3254	1 |- ( A <P B -> ( 2nd ` B ) C_ ( 2nd ` ( A +P. C ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   E.wex 1545   e. wcel 2209   E.wrex 2529   {crab 2532   C_ wss 3220   <.cop 3712   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   1stc1st 6372   2ndc2nd 6373   Q.cnq 7647   +Q cplq 7649   <Q cltq 7652   P.cnp 7658   +P. cpp 7660   <P cltp 7662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-2o 6688  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-pli 7672  df-mi 7673  df-lti 7674  df-plpq 7711  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-plqqs 7716  df-mqqs 7717  df-1nqqs 7718  df-rq 7719  df-ltnqqs 7720  df-enq0 7791  df-nq0 7792  df-0nq0 7793  df-plq0 7794  df-mq0 7795  df-inp 7833  df-iplp 7835  df-iltp 7837

This theorem is used by:  ltexpri  7980