Theorem addlocprlemgt 7594

Description: Lemma for addlocpr 7596. The ( D +Q E ) <Q Q case. (Contributed by Jim Kingdon, 6-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
addlocprlemgt	|- ( ph -> ( ( D +Q E ) <Q Q -> R e. ( 2nd ` ( A +P. B ) ) ) )

Proof of Theorem addlocprlemgt

Step	Hyp	Ref	Expression
1		addlocprlem.a	. . . . . . 7 |- ( ph -> A e. P. )
2		addlocprlem.b	. . . . . . 7 |- ( ph -> B e. P. )
3		addlocprlem.qr	. . . . . . 7 |- ( ph -> Q <Q R )
4		addlocprlem.p	. . . . . . 7 |- ( ph -> P e. Q. )
5		addlocprlem.qppr	. . . . . . 7 |- ( ph -> ( Q +Q ( P +Q P ) ) = R )
6		addlocprlem.dlo	. . . . . . 7 |- ( ph -> D e. ( 1st ` A ) )
7		addlocprlem.uup	. . . . . . 7 |- ( ph -> U e. ( 2nd ` A ) )
8		addlocprlem.du	. . . . . . 7 |- ( ph -> U <Q ( D +Q P ) )
9		addlocprlem.elo	. . . . . . 7 |- ( ph -> E e. ( 1st ` B ) )
10		addlocprlem.tup	. . . . . . 7 |- ( ph -> T e. ( 2nd ` B ) )
11		addlocprlem.et	. . . . . . 7 |- ( ph -> T <Q ( E +Q P ) )
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	addlocprlemeqgt 7592	. . . . . 6 |- ( ph -> ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) )
13	12	adantr 276	. . . . 5 |- ( ( ph /\ ( D +Q E ) <Q Q ) -> ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) )
14		prop 7535	. . . . . . . . . . . 12 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
15	1, 14	syl 14	. . . . . . . . . . 11 |- ( ph -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
16		elprnql 7541	. . . . . . . . . . 11 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ D e. ( 1st ` A ) ) -> D e. Q. )
17	15, 6, 16	syl2anc 411	. . . . . . . . . 10 |- ( ph -> D e. Q. )
18		prop 7535	. . . . . . . . . . . 12 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
19	2, 18	syl 14	. . . . . . . . . . 11 |- ( ph -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
20		elprnql 7541	. . . . . . . . . . 11 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ E e. ( 1st ` B ) ) -> E e. Q. )
21	19, 9, 20	syl2anc 411	. . . . . . . . . 10 |- ( ph -> E e. Q. )
22		addclnq 7435	. . . . . . . . . 10 |- ( ( D e. Q. /\ E e. Q. ) -> ( D +Q E ) e. Q. )
23	17, 21, 22	syl2anc 411	. . . . . . . . 9 |- ( ph -> ( D +Q E ) e. Q. )
24		ltrelnq 7425	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
25	24	brel 4711	. . . . . . . . . . 11 |- ( Q <Q R -> ( Q e. Q. /\ R e. Q. ) )
26	3, 25	syl 14	. . . . . . . . . 10 |- ( ph -> ( Q e. Q. /\ R e. Q. ) )
27	26	simpld 112	. . . . . . . . 9 |- ( ph -> Q e. Q. )
28		addclnq 7435	. . . . . . . . . 10 |- ( ( P e. Q. /\ P e. Q. ) -> ( P +Q P ) e. Q. )
29	4, 4, 28	syl2anc 411	. . . . . . . . 9 |- ( ph -> ( P +Q P ) e. Q. )
30		ltanqg 7460	. . . . . . . . 9 |- ( ( ( D +Q E ) e. Q. /\ Q e. Q. /\ ( P +Q P ) e. Q. ) -> ( ( D +Q E ) <Q Q <-> ( ( P +Q P ) +Q ( D +Q E ) ) <Q ( ( P +Q P ) +Q Q ) ) )
31	23, 27, 29, 30	syl3anc 1249	. . . . . . . 8 |- ( ph -> ( ( D +Q E ) <Q Q <-> ( ( P +Q P ) +Q ( D +Q E ) ) <Q ( ( P +Q P ) +Q Q ) ) )
32		addcomnqg 7441	. . . . . . . . . 10 |- ( ( ( P +Q P ) e. Q. /\ ( D +Q E ) e. Q. ) -> ( ( P +Q P ) +Q ( D +Q E ) ) = ( ( D +Q E ) +Q ( P +Q P ) ) )
33	29, 23, 32	syl2anc 411	. . . . . . . . 9 |- ( ph -> ( ( P +Q P ) +Q ( D +Q E ) ) = ( ( D +Q E ) +Q ( P +Q P ) ) )
34		addcomnqg 7441	. . . . . . . . . 10 |- ( ( ( P +Q P ) e. Q. /\ Q e. Q. ) -> ( ( P +Q P ) +Q Q ) = ( Q +Q ( P +Q P ) ) )
35	29, 27, 34	syl2anc 411	. . . . . . . . 9 |- ( ph -> ( ( P +Q P ) +Q Q ) = ( Q +Q ( P +Q P ) ) )
36	33, 35	breq12d 4042	. . . . . . . 8 |- ( ph -> ( ( ( P +Q P ) +Q ( D +Q E ) ) <Q ( ( P +Q P ) +Q Q ) <-> ( ( D +Q E ) +Q ( P +Q P ) ) <Q ( Q +Q ( P +Q P ) ) ) )
37	31, 36	bitrd 188	. . . . . . 7 |- ( ph -> ( ( D +Q E ) <Q Q <-> ( ( D +Q E ) +Q ( P +Q P ) ) <Q ( Q +Q ( P +Q P ) ) ) )
38	37	biimpa 296	. . . . . 6 |- ( ( ph /\ ( D +Q E ) <Q Q ) -> ( ( D +Q E ) +Q ( P +Q P ) ) <Q ( Q +Q ( P +Q P ) ) )
39	5	breq2d 4041	. . . . . . 7 |- ( ph -> ( ( ( D +Q E ) +Q ( P +Q P ) ) <Q ( Q +Q ( P +Q P ) ) <-> ( ( D +Q E ) +Q ( P +Q P ) ) <Q R ) )
40	39	adantr 276	. . . . . 6 |- ( ( ph /\ ( D +Q E ) <Q Q ) -> ( ( ( D +Q E ) +Q ( P +Q P ) ) <Q ( Q +Q ( P +Q P ) ) <-> ( ( D +Q E ) +Q ( P +Q P ) ) <Q R ) )
41	38, 40	mpbid 147	. . . . 5 |- ( ( ph /\ ( D +Q E ) <Q Q ) -> ( ( D +Q E ) +Q ( P +Q P ) ) <Q R )
42	13, 41	jca 306	. . . 4 |- ( ( ph /\ ( D +Q E ) <Q Q ) -> ( ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) /\ ( ( D +Q E ) +Q ( P +Q P ) ) <Q R ) )
43		ltsonq 7458	. . . . 5 |- <Q Or Q.
44	43, 24	sotri 5061	. . . 4 |- ( ( ( U +Q T ) <Q ( ( D +Q E ) +Q ( P +Q P ) ) /\ ( ( D +Q E ) +Q ( P +Q P ) ) <Q R ) -> ( U +Q T ) <Q R )
45	42, 44	syl 14	. . 3 |- ( ( ph /\ ( D +Q E ) <Q Q ) -> ( U +Q T ) <Q R )
46	1, 7	jca 306	. . . . 5 |- ( ph -> ( A e. P. /\ U e. ( 2nd ` A ) ) )
47	2, 10	jca 306	. . . . 5 |- ( ph -> ( B e. P. /\ T e. ( 2nd ` B ) ) )
48	26	simprd 114	. . . . 5 |- ( ph -> R e. Q. )
49		addnqpru 7590	. . . . 5 |- ( ( ( ( A e. P. /\ U e. ( 2nd ` A ) ) /\ ( B e. P. /\ T e. ( 2nd ` B ) ) ) /\ R e. Q. ) -> ( ( U +Q T ) <Q R -> R e. ( 2nd ` ( A +P. B ) ) ) )
50	46, 47, 48, 49	syl21anc 1248	. . . 4 |- ( ph -> ( ( U +Q T ) <Q R -> R e. ( 2nd ` ( A +P. B ) ) ) )
51	50	adantr 276	. . 3 |- ( ( ph /\ ( D +Q E ) <Q Q ) -> ( ( U +Q T ) <Q R -> R e. ( 2nd ` ( A +P. B ) ) ) )
52	45, 51	mpd 13	. 2 |- ( ( ph /\ ( D +Q E ) <Q Q ) -> R e. ( 2nd ` ( A +P. B ) ) )
53	52	ex 115	1 |- ( ph -> ( ( D +Q E ) <Q Q -> R e. ( 2nd ` ( A +P. B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   <.cop 3621   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   1stc1st 6191   2ndc2nd 6192   Q.cnq 7340   +Q cplq 7342   <Q cltq 7345   P.cnp 7351   +P. cpp 7353

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-inp 7526  df-iplp 7528

This theorem is referenced by:  addlocprlem  7595