Theorem ltexprlemru 7420

Description: Lemma for ltexpri 7421. 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 7313	. . . . . . . 8 |- <P C_ ( P. X. P. )
2	1	brel 4591	. . . . . . 7 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
3	2	simprd 113	. . . . . 6 |- ( A <P B -> B e. P. )
4		prop 7283	. . . . . 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 7297	. . . . 5 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ w e. ( 2nd ` B ) ) -> E. t e. ( 2nd ` B ) t <Q w )
7	5, 6	sylan 281	. . . 4 |- ( ( A <P B /\ w e. ( 2nd ` B ) ) -> E. t e. ( 2nd ` B ) t <Q w )
8		simprr 521	. . . . . 6 |- ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) -> t <Q w )
9		elprnqu 7290	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ t e. ( 2nd ` B ) ) -> t e. Q. )
10	5, 9	sylan 281	. . . . . . . 8 |- ( ( A <P B /\ t e. ( 2nd ` B ) ) -> t e. Q. )
11	10	ad2ant2r 500	. . . . . . 7 |- ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) -> t e. Q. )
12		elprnqu 7290	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ w e. ( 2nd ` B ) ) -> w e. Q. )
13	5, 12	sylan 281	. . . . . . . 8 |- ( ( A <P B /\ w e. ( 2nd ` B ) ) -> w e. Q. )
14	13	adantr 274	. . . . . . 7 |- ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ ( t e. ( 2nd ` B ) /\ t <Q w ) ) -> w e. Q. )
15		ltexnqq 7216	. . . . . . 7 |- ( ( t e. Q. /\ w e. Q. ) -> ( t <Q w <-> E. v e. Q. ( t +Q v ) = w ) )
16	11, 14, 15	syl2anc 408	. . . . . 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 7283	. . . . . . . . . 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 7311	. . . . . . . . 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 281	. . . . . . . 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 468	. . . . . . 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 500	. . . . . 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 522	. . . . . . . . . . . . 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 479	. . . . . . . . . . . 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 7314	. . . . . . . . . . . . . 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 274	. . . . . . . . . . . . . 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 479	. . . . . . . . . . . . 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 524	. . . . . . . . . . . . . 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 274	. . . . . . . . . . . . 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 528	. . . . . . . . . . . . 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 7295	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) -> z <Q q )
37	20, 36	syl3an1 1249	. . . . . . . . . . . . 13 |- ( ( A <P B /\ z e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) -> z <Q q )
38	32, 34, 35, 37	syl3anc 1216	. . . . . . . . . . . 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 529	. . . . . . . . . . . . 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 524	. . . . . . . . . . . . . . 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 274	. . . . . . . . . . . . . 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 479	. . . . . . . . . . . . 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 7295	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ q e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) -> q <Q t )
44	5, 43	syl3an1 1249	. . . . . . . . . . . . 13 |- ( ( A <P B /\ q e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) -> q <Q t )
45	32, 39, 42, 44	syl3anc 1216	. . . . . . . . . . . 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 7206	. . . . . . . . . . . . 13 |- <Q Or Q.
47		ltrelnq 7173	. . . . . . . . . . . . 13 |- <Q C_ ( Q. X. Q. )
48	46, 47	sotri 4934	. . . . . . . . . . . 12 |- ( ( z <Q q /\ q <Q t ) -> z <Q t )
49	38, 45, 48	syl2anc 408	. . . . . . . . . . 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 2554	. . . . . . . . . 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 7217	. . . . . . . . . 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 525	. . . . . . . . . . . 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 479	. . . . . . . . . . 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 521	. . . . . . . . . . . 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 5781	. . . . . . . . . . . . 13 |- ( ( z +Q s ) = t -> ( ( z +Q s ) +Q v ) = ( t +Q v ) )
57	56	eqeq1d 2148	. . . . . . . . . . . 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 7289	. . . . . . . . . . . . . . . . 17 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
61	20, 60	sylan 281	. . . . . . . . . . . . . . . 16 |- ( ( A <P B /\ z e. ( 1st ` A ) ) -> z e. Q. )
62	61	adantlr 468	. . . . . . . . . . . . . . 15 |- ( ( ( A <P B /\ w e. ( 2nd ` B ) ) /\ z e. ( 1st ` A ) ) -> z e. Q. )
63	62	ad2ant2r 500	. . . . . . . . . . . . . 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 468	. . . . . . . . . . . . 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 479	. . . . . . . . . . . 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 524	. . . . . . . . . . . . 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 479	. . . . . . . . . . . 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 520	. . . . . . . . . . . 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 7189	. . . . . . . . . . . . 13 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
70	69	adantl 275	. . . . . . . . . . . 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 7190	. . . . . . . . . . . . 13 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( ( f +Q g ) +Q h ) = ( f +Q ( g +Q h ) ) )
72	71	adantl 275	. . . . . . . . . . . 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 5951	. . . . . . . . . . 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 525	. . . . . . . . . . . . . . 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 7293	. . . . . . . . . . . . . . . 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 281	. . . . . . . . . . . . . . 15 |- ( ( A <P B /\ u e. ( 2nd ` A ) ) -> ( u <Q ( z +Q v ) -> ( z +Q v ) e. ( 2nd ` A ) ) )
78	26, 75, 77	syl2anc 408	. . . . . . . . . . . . . 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 274	. . . . . . . . . . . 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 274	. . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . 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 2216	. . . . . . . . . . . . . 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 2202	. . . . . . . . . . . . . . . . 17 |- ( y = z -> ( y e. ( 1st ` A ) <-> z e. ( 1st ` A ) ) )
85		oveq1 5781	. . . . . . . . . . . . . . . . . 18 |- ( y = z -> ( y +Q s ) = ( z +Q s ) )
86	85	eleq1d 2208	. . . . . . . . . . . . . . . . 17 |- ( y = z -> ( ( y +Q s ) e. ( 2nd ` B ) <-> ( z +Q s ) e. ( 2nd ` B ) ) )
87	84, 86	anbi12d 464	. . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . 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 568	. . . . . . . . . . . . . 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 408	. . . . . . . . . . . . 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 7407	. . . . . . . . . . . . 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 413	. . . . . . . . . . . 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 479	. . . . . . . . . . . . . 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 7416	. . . . . . . . . . . . . 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 7276	. . . . . . . . . . . . . 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 7183	. . . . . . . . . . . . . 14 |- ( ( f e. Q. /\ v e. Q. ) -> ( f +Q v ) e. Q. )
100	98, 99	genppreclu 7323	. . . . . . . . . . . . 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 408	. . . . . . . . . . . 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 429	. . . . . . . . . . 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 2217	. . . . . . . . . 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 2217	. . . . . . . . 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 2554	. . . . . . . 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 2557	. . . . . 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 2554	. . . 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 2554	. . 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 3103	1 |- ( A <P B -> ( 2nd ` B ) C_ ( 2nd ` ( A +P. C ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   E.wex 1468   e. wcel 1480   E.wrex 2417   {crab 2420   C_ wss 3071   <.cop 3530   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   1stc1st 6036   2ndc2nd 6037   Q.cnq 7088   +Q cplq 7090   <Q cltq 7093   P.cnp 7099   +P. cpp 7101   <P cltp 7103

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-iplp 7276  df-iltp 7278

This theorem is referenced by:  ltexpri  7421