Theorem ltexprlemru 7368

Description: Lemma for ltexpri 7369. 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 7261	. . . . . . . 8 |- <P C_ ( P. X. P. )
2	1	brel 4551	. . . . . . 7 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
3	2	simprd 113	. . . . . 6 |- ( A <P B -> B e. P. )
4		prop 7231	. . . . . 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 7245	. . . . 5 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ w e. ( 2nd ` B ) ) -> E. t e. ( 2nd ` B ) t <Q w )
7	5, 6	sylan 279	. . . 4 |- ( ( A <P B /\ w e. ( 2nd ` B ) ) -> E. t e. ( 2nd ` B ) t <Q w )
8		simprr 504	. . . . . 6 |- ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) -> t <Q w )
9		elprnqu 7238	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ t e. ( 2nd ` B ) ) -> t e. Q. )
10	5, 9	sylan 279	. . . . . . . 8 |- ( ( A <P B /\ t e. ( 2nd ` B ) ) -> t e. Q. )
11	10	ad2ant2r 498	. . . . . . 7 |- ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) -> t e. Q. )
12		elprnqu 7238	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ w e. ( 2nd ` B ) ) -> w e. Q. )
13	5, 12	sylan 279	. . . . . . . 8 |- ( ( A <P B /\ w e. ( 2nd ` B ) ) -> w e. Q. )
14	13	adantr 272	. . . . . . 7 |- ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) -> w e. Q. )
15		ltexnqq 7164	. . . . . . 7 |- ( ( t e. Q. /\ w e. Q. ) -> ( t <Q w <-> E. v e. Q. ( t +Q v ) = w ) )
16	11, 14, 15	syl2anc 406	. . . . . 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 146	. . . . 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 111	. . . . . . . . . 10 |- ( A <P B -> A e. P. )
19		prop 7231	. . . . . . . . . 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 7259	. . . . . . . . 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 279	. . . . . . . 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 466	. . . . . . 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 498	. . . . . 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 505	. . . . . . . . . . . . 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 477	. . . . . . . . . . . 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 7262	. . . . . . . . . . . . . 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 143	. . . . . . . . . . . . 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 272	. . . . . . . . . . . . . 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 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 ) ) ) /\ u <Q ( z +Q v ) ) /\ ( q e. Q. /\ ( q e. ( 2nd ` A ) /\ q e. ( 1st ` B ) ) ) ) -> A <P B )
33		simplrl 507	. . . . . . . . . . . . . 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 272	. . . . . . . . . . . . 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 511	. . . . . . . . . . . . 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 7243	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) -> z <Q q )
37	20, 36	syl3an1 1232	. . . . . . . . . . . . 13 |- ( ( A <P B /\ z e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) -> z <Q q )
38	32, 34, 35, 37	syl3anc 1199	. . . . . . . . . . . 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 512	. . . . . . . . . . . . 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 507	. . . . . . . . . . . . . . 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 272	. . . . . . . . . . . . . 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 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 ) ) ) /\ u <Q ( z +Q v ) ) /\ ( q e. Q. /\ ( q e. ( 2nd ` A ) /\ q e. ( 1st ` B ) ) ) ) -> t e. ( 2nd ` B ) )
43		prltlu 7243	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ q e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) -> q <Q t )
44	5, 43	syl3an1 1232	. . . . . . . . . . . . 13 |- ( ( A <P B /\ q e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) -> q <Q t )
45	32, 39, 42, 44	syl3anc 1199	. . . . . . . . . . . 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 7154	. . . . . . . . . . . . 13 |- <Q Or Q.
47		ltrelnq 7121	. . . . . . . . . . . . 13 |- <Q C_ ( Q. X. Q. )
48	46, 47	sotri 4892	. . . . . . . . . . . 12 |- ( ( z <Q q /\ q <Q t ) -> z <Q t )
49	38, 45, 48	syl2anc 406	. . . . . . . . . . 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 2528	. . . . . . . . . 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 7165	. . . . . . . . . 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 508	. . . . . . . . . . . 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 477	. . . . . . . . . . 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 504	. . . . . . . . . . . 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 5735	. . . . . . . . . . . . 13 |- ( ( z +Q s ) = t -> ( ( z +Q s ) +Q v ) = ( t +Q v ) )
57	56	eqeq1d 2123	. . . . . . . . . . . 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 166	. . . . . . . . . 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 7237	. . . . . . . . . . . . . . . . 17 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
61	20, 60	sylan 279	. . . . . . . . . . . . . . . 16 |- ( ( A <P B /\ z e. ( 1st ` A ) ) -> z e. Q. )
62	61	adantlr 466	. . . . . . . . . . . . . . 15 |- ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ z e. ( 1st ` A ) ) -> z e. Q. )
63	62	ad2ant2r 498	. . . . . . . . . . . . . 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 466	. . . . . . . . . . . . 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 477	. . . . . . . . . . . 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 507	. . . . . . . . . . . . 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 477	. . . . . . . . . . . 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 503	. . . . . . . . . . . 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 7137	. . . . . . . . . . . . 13 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
70	69	adantl 273	. . . . . . . . . . . 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 7138	. . . . . . . . . . . . 13 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( ( f +Q g ) +Q h ) = ( f +Q ( g +Q h ) ) )
72	71	adantl 273	. . . . . . . . . . . 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 5905	. . . . . . . . . . 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 109	. . . . . . . . . . . . . 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 508	. . . . . . . . . . . . . . 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 7241	. . . . . . . . . . . . . . . 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 279	. . . . . . . . . . . . . . 15 |- ( ( A <P B /\ u e. ( 2nd ` A ) ) -> ( u <Q ( z +Q v ) -> ( z +Q v ) e. ( 2nd ` A ) ) )
78	26, 75, 77	syl2anc 406	. . . . . . . . . . . . . 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 272	. . . . . . . . . . . 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 272	. . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . 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 2191	. . . . . . . . . . . . . 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 2177	. . . . . . . . . . . . . . . . 17 |- ( y = z -> ( y e. ( 1st ` A ) <-> z e. ( 1st ` A ) ) )
85		oveq1 5735	. . . . . . . . . . . . . . . . . 18 |- ( y = z -> ( y +Q s ) = ( z +Q s ) )
86	85	eleq1d 2183	. . . . . . . . . . . . . . . . 17 |- ( y = z -> ( ( y +Q s ) e. ( 2nd ` B ) <-> ( z +Q s ) e. ( 2nd ` B ) ) )
87	84, 86	anbi12d 462	. . . . . . . . . . . . . . . 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 2745	. . . . . . . . . . . . . . 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 551	. . . . . . . . . . . . . 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 406	. . . . . . . . . . . . 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 7355	. . . . . . . . . . . . 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 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 ) ) -> s e. ( 2nd ` C ) )
94	31	ad2antrr 477	. . . . . . . . . . . . . 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 7364	. . . . . . . . . . . . . 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 7224	. . . . . . . . . . . . . 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 7131	. . . . . . . . . . . . . 14 |- ( ( f e. Q. /\ v e. Q. ) -> ( f +Q v ) e. Q. )
100	98, 99	genppreclu 7271	. . . . . . . . . . . . 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 406	. . . . . . . . . . . 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 427	. . . . . . . . . . 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 2192	. . . . . . . . . 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 2192	. . . . . . . . 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 2528	. . . . . . . 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 114	. . . . . . 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 2531	. . . . . 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 2528	. . . 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 2528	. . 3 |- ( ( A <P B /\ w e. ( 2nd ` B ) ) -> w e. ( 2nd ` ( A +P. C ) ) )
111	110	ex 114	. 2 |- ( A <P B -> ( w e. ( 2nd ` B ) -> w e. ( 2nd ` ( A +P. C ) ) ) )
112	111	ssrdv 3069	1 |- ( A <P B -> ( 2nd ` B ) C_ ( 2nd ` ( A +P. C ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 945   = wceq 1314   E.wex 1451   e. wcel 1463   E.wrex 2391   {crab 2394   C_ wss 3037   <.cop 3496   class class class wbr 3895   ` cfv 5081  (class class class)co 5728   1stc1st 5990   2ndc2nd 5991   Q.cnq 7036   +Q cplq 7038   <Q cltq 7041   P.cnp 7047   +P. cpp 7049   <P cltp 7051

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-eprel 4171  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-irdg 6221  df-1o 6267  df-2o 6268  df-oadd 6271  df-omul 6272  df-er 6383  df-ec 6385  df-qs 6389  df-ni 7060  df-pli 7061  df-mi 7062  df-lti 7063  df-plpq 7100  df-mpq 7101  df-enq 7103  df-nqqs 7104  df-plqqs 7105  df-mqqs 7106  df-1nqqs 7107  df-rq 7108  df-ltnqqs 7109  df-enq0 7180  df-nq0 7181  df-0nq0 7182  df-plq0 7183  df-mq0 7184  df-inp 7222  df-iplp 7224  df-iltp 7226

This theorem is referenced by:  ltexpri  7369