Theorem mulnqprlemfu 7696

Description: Lemma for mulnqpr 7697. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)

Assertion

Ref	Expression
mulnqprlemfu	|- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) C_ ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )

Distinct variable groups:   A, l, u   B, l, u

Proof of Theorem mulnqprlemfu

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulnqprlemrl 7693	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. ) -> ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) C_ ( 1st ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) )
2		ltsonq 7518	. . . . . . . . 9 |- <Q Or Q.
3		mulclnq 7496	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. ) -> ( A .Q B ) e. Q. )
4		sonr 4368	. . . . . . . . 9 |- ( ( <Q Or Q. /\ ( A .Q B ) e. Q. ) -> -. ( A .Q B ) <Q ( A .Q B ) )
5	2, 3, 4	sylancr 414	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. ) -> -. ( A .Q B ) <Q ( A .Q B ) )
6		ltrelnq 7485	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
7	6	brel 4731	. . . . . . . . . . 11 |- ( ( A .Q B ) <Q ( A .Q B ) -> ( ( A .Q B ) e. Q. /\ ( A .Q B ) e. Q. ) )
8	7	simpld 112	. . . . . . . . . 10 |- ( ( A .Q B ) <Q ( A .Q B ) -> ( A .Q B ) e. Q. )
9		elex 2784	. . . . . . . . . 10 |- ( ( A .Q B ) e. Q. -> ( A .Q B ) e. _V )
10	8, 9	syl 14	. . . . . . . . 9 |- ( ( A .Q B ) <Q ( A .Q B ) -> ( A .Q B ) e. _V )
11		breq1 4050	. . . . . . . . 9 |- ( l = ( A .Q B ) -> ( l <Q ( A .Q B ) <-> ( A .Q B ) <Q ( A .Q B ) ) )
12	10, 11	elab3 2926	. . . . . . . 8 |- ( ( A .Q B ) e. { l | l <Q ( A .Q B ) } <-> ( A .Q B ) <Q ( A .Q B ) )
13	5, 12	sylnibr 679	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. ) -> -. ( A .Q B ) e. { l | l <Q ( A .Q B ) } )
14		ltnqex 7669	. . . . . . . . 9 |- { l | l <Q ( A .Q B ) } e. _V
15		gtnqex 7670	. . . . . . . . 9 |- { u | ( A .Q B ) <Q u } e. _V
16	14, 15	op1st 6239	. . . . . . . 8 |- ( 1st ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) = { l | l <Q ( A .Q B ) }
17	16	eleq2i 2273	. . . . . . 7 |- ( ( A .Q B ) e. ( 1st ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) <-> ( A .Q B ) e. { l | l <Q ( A .Q B ) } )
18	13, 17	sylnibr 679	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. ) -> -. ( A .Q B ) e. ( 1st ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) )
19	1, 18	ssneldd 3197	. . . . 5 |- ( ( A e. Q. /\ B e. Q. ) -> -. ( A .Q B ) e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )
20	19	adantr 276	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) ) -> -. ( A .Q B ) e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )
21		nqprlu 7667	. . . . . . . 8 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
22		nqprlu 7667	. . . . . . . 8 |- ( B e. Q. -> <. { l | l <Q B } , { u | B <Q u } >. e. P. )
23		mulclpr 7692	. . . . . . . 8 |- ( ( <. { l | l <Q A } , { u | A <Q u } >. e. P. /\ <. { l | l <Q B } , { u | B <Q u } >. e. P. ) -> ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) e. P. )
24	21, 22, 23	syl2an 289	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. ) -> ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) e. P. )
25		prop 7595	. . . . . . 7 |- ( ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) e. P. -> <. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) , ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) >. e. P. )
26	24, 25	syl 14	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. ) -> <. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) , ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) >. e. P. )
27		vex 2776	. . . . . . . 8 |- r e. _V
28		breq2 4051	. . . . . . . 8 |- ( u = r -> ( ( A .Q B ) <Q u <-> ( A .Q B ) <Q r ) )
29	14, 15	op2nd 6240	. . . . . . . 8 |- ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) = { u | ( A .Q B ) <Q u }
30	27, 28, 29	elab2 2922	. . . . . . 7 |- ( r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) <-> ( A .Q B ) <Q r )
31	30	biimpi 120	. . . . . 6 |- ( r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) -> ( A .Q B ) <Q r )
32		prloc 7611	. . . . . 6 |- ( ( <. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) , ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) >. e. P. /\ ( A .Q B ) <Q r ) -> ( ( A .Q B ) e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) \/ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) )
33	26, 31, 32	syl2an 289	. . . . 5 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) ) -> ( ( A .Q B ) e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) \/ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) )
34	33	orcomd 731	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) ) -> ( r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) \/ ( A .Q B ) e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) )
35	20, 34	ecased 1362	. . 3 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) ) -> r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )
36	35	ex 115	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) -> r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) )
37	36	ssrdv 3200	1 |- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) C_ ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 710   e. wcel 2177   {cab 2192   _Vcvv 2773   C_ wss 3167   <.cop 3637   class class class wbr 4047   Or wor 4346   ` cfv 5276  (class class class)co 5951   1stc1st 6231   2ndc2nd 6232   Q.cnq 7400   .Q cmq 7403   <Q cltq 7405   P.cnp 7411   .P. cmp 7414

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-eprel 4340  df-id 4344  df-po 4347  df-iso 4348  df-iord 4417  df-on 4419  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-irdg 6463  df-1o 6509  df-2o 6510  df-oadd 6513  df-omul 6514  df-er 6627  df-ec 6629  df-qs 6633  df-ni 7424  df-pli 7425  df-mi 7426  df-lti 7427  df-plpq 7464  df-mpq 7465  df-enq 7467  df-nqqs 7468  df-plqqs 7469  df-mqqs 7470  df-1nqqs 7471  df-rq 7472  df-ltnqqs 7473  df-enq0 7544  df-nq0 7545  df-0nq0 7546  df-plq0 7547  df-mq0 7548  df-inp 7586  df-imp 7589

This theorem is referenced by:  mulnqpr  7697