Theorem addclprlem2 9839

Description: Lemma to prove downward closure in positive real addition. Part of proof of Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
addclprlem2	|- ( ( ( ( A e. P. /\ g e. A ) /\ ( B e. P. /\ h e. B ) ) /\ x e. Q. ) -> ( x <Q ( g +Q h ) -> x e. ( A +P. B ) ) )

Distinct variable groups:   x, g, h   x, A   x, B

Allowed substitution hints:   A( g, h)   B( g, h)

Proof of Theorem addclprlem2

Dummy variables y z w v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addclprlem1 9838	. . . . 5 |- ( ( ( A e. P. /\ g e. A ) /\ x e. Q. ) -> ( x <Q ( g +Q h ) -> ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q g ) e. A ) )
2	1	adantlr 751	. . . 4 |- ( ( ( ( A e. P. /\ g e. A ) /\ ( B e. P. /\ h e. B ) ) /\ x e. Q. ) -> ( x <Q ( g +Q h ) -> ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q g ) e. A ) )
3		addclprlem1 9838	. . . . . 6 |- ( ( ( B e. P. /\ h e. B ) /\ x e. Q. ) -> ( x <Q ( h +Q g ) -> ( ( x .Q ( *Q ` ( h +Q g ) ) ) .Q h ) e. B ) )
4		addcomnq 9773	. . . . . . 7 |- ( g +Q h ) = ( h +Q g )
5	4	breq2i 4661	. . . . . 6 |- ( x <Q ( g +Q h ) <-> x <Q ( h +Q g ) )
6	4	fveq2i 6194	. . . . . . . . 9 |- ( *Q ` ( g +Q h ) ) = ( *Q ` ( h +Q g ) )
7	6	oveq2i 6661	. . . . . . . 8 |- ( x .Q ( *Q ` ( g +Q h ) ) ) = ( x .Q ( *Q ` ( h +Q g ) ) )
8	7	oveq1i 6660	. . . . . . 7 |- ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q h ) = ( ( x .Q ( *Q ` ( h +Q g ) ) ) .Q h )
9	8	eleq1i 2692	. . . . . 6 |- ( ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q h ) e. B <-> ( ( x .Q ( *Q ` ( h +Q g ) ) ) .Q h ) e. B )
10	3, 5, 9	3imtr4g 285	. . . . 5 |- ( ( ( B e. P. /\ h e. B ) /\ x e. Q. ) -> ( x <Q ( g +Q h ) -> ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q h ) e. B ) )
11	10	adantll 750	. . . 4 |- ( ( ( ( A e. P. /\ g e. A ) /\ ( B e. P. /\ h e. B ) ) /\ x e. Q. ) -> ( x <Q ( g +Q h ) -> ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q h ) e. B ) )
12	2, 11	jcad 555	. . 3 |- ( ( ( ( A e. P. /\ g e. A ) /\ ( B e. P. /\ h e. B ) ) /\ x e. Q. ) -> ( x <Q ( g +Q h ) -> ( ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q g ) e. A /\ ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q h ) e. B ) ) )
13		simpl 473	. . . 4 |- ( ( ( ( A e. P. /\ g e. A ) /\ ( B e. P. /\ h e. B ) ) /\ x e. Q. ) -> ( ( A e. P. /\ g e. A ) /\ ( B e. P. /\ h e. B ) ) )
14		simpl 473	. . . . 5 |- ( ( A e. P. /\ g e. A ) -> A e. P. )
15		simpl 473	. . . . 5 |- ( ( B e. P. /\ h e. B ) -> B e. P. )
16	14, 15	anim12i 590	. . . 4 |- ( ( ( A e. P. /\ g e. A ) /\ ( B e. P. /\ h e. B ) ) -> ( A e. P. /\ B e. P. ) )
17		df-plp 9805	. . . . 5 |- +P. = ( w e. P. , v e. P. |-> { x | E. y e. w E. z e. v x = ( y +Q z ) } )
18		addclnq 9767	. . . . 5 |- ( ( y e. Q. /\ z e. Q. ) -> ( y +Q z ) e. Q. )
19	17, 18	genpprecl 9823	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( ( ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q g ) e. A /\ ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q h ) e. B ) -> ( ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q g ) +Q ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q h ) ) e. ( A +P. B ) ) )
20	13, 16, 19	3syl 18	. . 3 |- ( ( ( ( A e. P. /\ g e. A ) /\ ( B e. P. /\ h e. B ) ) /\ x e. Q. ) -> ( ( ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q g ) e. A /\ ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q h ) e. B ) -> ( ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q g ) +Q ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q h ) ) e. ( A +P. B ) ) )
21	12, 20	syld 47	. 2 |- ( ( ( ( A e. P. /\ g e. A ) /\ ( B e. P. /\ h e. B ) ) /\ x e. Q. ) -> ( x <Q ( g +Q h ) -> ( ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q g ) +Q ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q h ) ) e. ( A +P. B ) ) )
22		distrnq 9783	. . . . 5 |- ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q ( g +Q h ) ) = ( ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q g ) +Q ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q h ) )
23		mulassnq 9781	. . . . 5 |- ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q ( g +Q h ) ) = ( x .Q ( ( *Q ` ( g +Q h ) ) .Q ( g +Q h ) ) )
24	22, 23	eqtr3i 2646	. . . 4 |- ( ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q g ) +Q ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q h ) ) = ( x .Q ( ( *Q ` ( g +Q h ) ) .Q ( g +Q h ) ) )
25		mulcomnq 9775	. . . . . . 7 |- ( ( *Q ` ( g +Q h ) ) .Q ( g +Q h ) ) = ( ( g +Q h ) .Q ( *Q ` ( g +Q h ) ) )
26		elprnq 9813	. . . . . . . . 9 |- ( ( A e. P. /\ g e. A ) -> g e. Q. )
27		elprnq 9813	. . . . . . . . 9 |- ( ( B e. P. /\ h e. B ) -> h e. Q. )
28	26, 27	anim12i 590	. . . . . . . 8 |- ( ( ( A e. P. /\ g e. A ) /\ ( B e. P. /\ h e. B ) ) -> ( g e. Q. /\ h e. Q. ) )
29		addclnq 9767	. . . . . . . 8 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
30		recidnq 9787	. . . . . . . 8 |- ( ( g +Q h ) e. Q. -> ( ( g +Q h ) .Q ( *Q ` ( g +Q h ) ) ) = 1Q )
31	28, 29, 30	3syl 18	. . . . . . 7 |- ( ( ( A e. P. /\ g e. A ) /\ ( B e. P. /\ h e. B ) ) -> ( ( g +Q h ) .Q ( *Q ` ( g +Q h ) ) ) = 1Q )
32	25, 31	syl5eq 2668	. . . . . 6 |- ( ( ( A e. P. /\ g e. A ) /\ ( B e. P. /\ h e. B ) ) -> ( ( *Q ` ( g +Q h ) ) .Q ( g +Q h ) ) = 1Q )
33	32	oveq2d 6666	. . . . 5 |- ( ( ( A e. P. /\ g e. A ) /\ ( B e. P. /\ h e. B ) ) -> ( x .Q ( ( *Q ` ( g +Q h ) ) .Q ( g +Q h ) ) ) = ( x .Q 1Q ) )
34		mulidnq 9785	. . . . 5 |- ( x e. Q. -> ( x .Q 1Q ) = x )
35	33, 34	sylan9eq 2676	. . . 4 |- ( ( ( ( A e. P. /\ g e. A ) /\ ( B e. P. /\ h e. B ) ) /\ x e. Q. ) -> ( x .Q ( ( *Q ` ( g +Q h ) ) .Q ( g +Q h ) ) ) = x )
36	24, 35	syl5eq 2668	. . 3 |- ( ( ( ( A e. P. /\ g e. A ) /\ ( B e. P. /\ h e. B ) ) /\ x e. Q. ) -> ( ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q g ) +Q ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q h ) ) = x )
37	36	eleq1d 2686	. 2 |- ( ( ( ( A e. P. /\ g e. A ) /\ ( B e. P. /\ h e. B ) ) /\ x e. Q. ) -> ( ( ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q g ) +Q ( ( x .Q ( *Q ` ( g +Q h ) ) ) .Q h ) ) e. ( A +P. B ) <-> x e. ( A +P. B ) ) )
38	21, 37	sylibd 229	1 |- ( ( ( ( A e. P. /\ g e. A ) /\ ( B e. P. /\ h e. B ) ) /\ x e. Q. ) -> ( x <Q ( g +Q h ) -> x e. ( A +P. B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Q.cnq 9674   1Qc1q 9675   +Q cplq 9677   .Q cmq 9678   *Qcrq 9679   <Q cltq 9680   P.cnp 9681   +P. cpp 9683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-rq 9739  df-ltnq 9740  df-np 9803  df-plp 9805

This theorem is referenced by:  addclpr  9840