Theorem addlocprlemeqgt 7647

Description: Lemma for addlocpr 7651. This is a step used in both the Q = ( D +Q E ) and ( D +Q E ) <Q Q cases. (Contributed by Jim Kingdon, 7-Dec-2019.)

Hypotheses

Ref	Expression
addlocprlem.a	|- ( ph -> A e. P. )
addlocprlem.b	|- ( ph -> B e. P. )
addlocprlem.qr	|- ( ph -> Q <Q R )
addlocprlem.p	|- ( ph -> P e. Q. )
addlocprlem.qppr	|- ( ph -> ( Q +Q ( P +Q P ) ) = R )
addlocprlem.dlo	|- ( ph -> D e. ( 1st ` A ) )
addlocprlem.uup	|- ( ph -> U e. ( 2nd ` A ) )
addlocprlem.du	|- ( ph -> U <Q ( D +Q P ) )
addlocprlem.elo	|- ( ph -> E e. ( 1st ` B ) )
addlocprlem.tup	|- ( ph -> T e. ( 2nd ` B ) )
addlocprlem.et	|- ( ph -> T <Q ( E +Q P ) )

Assertion

Ref	Expression
addlocprlemeqgt	|- ( ph -> ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) )

Proof of Theorem addlocprlemeqgt

Dummy variables f g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addlocprlem.du	. . 3 |- ( ph -> U <Q ( D +Q P ) )
2		addlocprlem.et	. . 3 |- ( ph -> T <Q ( E +Q P ) )
3		addlocprlem.a	. . . . . 6 |- ( ph -> A e. P. )
4		prop 7590	. . . . . 6 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
5	3, 4	syl 14	. . . . 5 |- ( ph -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
6		addlocprlem.uup	. . . . 5 |- ( ph -> U e. ( 2nd ` A ) )
7		elprnqu 7597	. . . . 5 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ U e. ( 2nd ` A ) ) -> U e. Q. )
8	5, 6, 7	syl2anc 411	. . . 4 |- ( ph -> U e. Q. )
9		addlocprlem.dlo	. . . . . 6 |- ( ph -> D e. ( 1st ` A ) )
10		elprnql 7596	. . . . . 6 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ D e. ( 1st ` A ) ) -> D e. Q. )
11	5, 9, 10	syl2anc 411	. . . . 5 |- ( ph -> D e. Q. )
12		addlocprlem.p	. . . . 5 |- ( ph -> P e. Q. )
13		addclnq 7490	. . . . 5 |- ( ( D e. Q. /\ P e. Q. ) -> ( D +Q P ) e. Q. )
14	11, 12, 13	syl2anc 411	. . . 4 |- ( ph -> ( D +Q P ) e. Q. )
15		addlocprlem.b	. . . . . 6 |- ( ph -> B e. P. )
16		prop 7590	. . . . . 6 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
17	15, 16	syl 14	. . . . 5 |- ( ph -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
18		addlocprlem.tup	. . . . 5 |- ( ph -> T e. ( 2nd ` B ) )
19		elprnqu 7597	. . . . 5 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ T e. ( 2nd ` B ) ) -> T e. Q. )
20	17, 18, 19	syl2anc 411	. . . 4 |- ( ph -> T e. Q. )
21		addlocprlem.elo	. . . . . 6 |- ( ph -> E e. ( 1st ` B ) )
22		elprnql 7596	. . . . . 6 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ E e. ( 1st ` B ) ) -> E e. Q. )
23	17, 21, 22	syl2anc 411	. . . . 5 |- ( ph -> E e. Q. )
24		addclnq 7490	. . . . 5 |- ( ( E e. Q. /\ P e. Q. ) -> ( E +Q P ) e. Q. )
25	23, 12, 24	syl2anc 411	. . . 4 |- ( ph -> ( E +Q P ) e. Q. )
26		lt2addnq 7519	. . . 4 |- ( ( ( U e. Q. /\ ( D +Q P ) e. Q. ) /\ ( T e. Q. /\ ( E +Q P ) e. Q. ) ) -> ( ( U <Q ( D +Q P ) /\ T <Q ( E +Q P ) ) -> ( U +Q T ) <Q ( ( D +Q P ) +Q ( E +Q P ) ) ) )
27	8, 14, 20, 25, 26	syl22anc 1251	. . 3 |- ( ph -> ( ( U <Q ( D +Q P ) /\ T <Q ( E +Q P ) ) -> ( U +Q T ) <Q ( ( D +Q P ) +Q ( E +Q P ) ) ) )
28	1, 2, 27	mp2and 433	. 2 |- ( ph -> ( U +Q T ) <Q ( ( D +Q P ) +Q ( E +Q P ) ) )
29		addcomnqg 7496	. . . 4 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
30	29	adantl 277	. . 3 |- ( ( ph /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
31		addassnqg 7497	. . . 4 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( ( f +Q g ) +Q h ) = ( f +Q ( g +Q h ) ) )
32	31	adantl 277	. . 3 |- ( ( ph /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( ( f +Q g ) +Q h ) = ( f +Q ( g +Q h ) ) )
33		addclnq 7490	. . . 4 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) e. Q. )
34	33	adantl 277	. . 3 |- ( ( ph /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) e. Q. )
35	11, 12, 23, 30, 32, 12, 34	caov4d 6133	. 2 |- ( ph -> ( ( D +Q P ) +Q ( E +Q P ) ) = ( ( D +Q E ) +Q ( P +Q P ) ) )
36	28, 35	breqtrd 4071	1 |- ( ph -> ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2176   <.cop 3636   class class class wbr 4045   ` cfv 5272  (class class class)co 5946   1stc1st 6226   2ndc2nd 6227   Q.cnq 7395   +Q cplq 7397   <Q cltq 7400   P.cnp 7406

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637

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 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-eprel 4337  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-irdg 6458  df-oadd 6508  df-omul 6509  df-er 6622  df-ec 6624  df-qs 6628  df-ni 7419  df-pli 7420  df-mi 7421  df-lti 7422  df-plpq 7459  df-enq 7462  df-nqqs 7463  df-plqqs 7464  df-ltnqqs 7468  df-inp 7581

This theorem is referenced by:  addlocprlemeq  7648  addlocprlemgt  7649