Theorem ltaprlem 7680

Description: Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.)

Assertion

Ref	Expression
ltaprlem	|- ( C e. P. -> ( A <P B -> ( C +P. A ) <P ( C +P. B ) ) )

Proof of Theorem ltaprlem

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltexpri 7675	. . . 4 |- ( A <P B -> E. x e. P. ( A +P. x ) = B )
2	1	adantr 276	. . 3 |- ( ( A <P B /\ C e. P. ) -> E. x e. P. ( A +P. x ) = B )
3		simplr 528	. . . . . 6 |- ( ( ( A <P B /\ C e. P. ) /\ ( x e. P. /\ ( A +P. x ) = B ) ) -> C e. P. )
4		ltrelpr 7567	. . . . . . . . . 10 |- <P C_ ( P. X. P. )
5	4	brel 4712	. . . . . . . . 9 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
6	5	simpld 112	. . . . . . . 8 |- ( A <P B -> A e. P. )
7	6	adantr 276	. . . . . . 7 |- ( ( A <P B /\ C e. P. ) -> A e. P. )
8	7	adantr 276	. . . . . 6 |- ( ( ( A <P B /\ C e. P. ) /\ ( x e. P. /\ ( A +P. x ) = B ) ) -> A e. P. )
9		addclpr 7599	. . . . . 6 |- ( ( C e. P. /\ A e. P. ) -> ( C +P. A ) e. P. )
10	3, 8, 9	syl2anc 411	. . . . 5 |- ( ( ( A <P B /\ C e. P. ) /\ ( x e. P. /\ ( A +P. x ) = B ) ) -> ( C +P. A ) e. P. )
11		simprl 529	. . . . 5 |- ( ( ( A <P B /\ C e. P. ) /\ ( x e. P. /\ ( A +P. x ) = B ) ) -> x e. P. )
12		ltaddpr 7659	. . . . 5 |- ( ( ( C +P. A ) e. P. /\ x e. P. ) -> ( C +P. A ) <P ( ( C +P. A ) +P. x ) )
13	10, 11, 12	syl2anc 411	. . . 4 |- ( ( ( A <P B /\ C e. P. ) /\ ( x e. P. /\ ( A +P. x ) = B ) ) -> ( C +P. A ) <P ( ( C +P. A ) +P. x ) )
14		addassprg 7641	. . . . . 6 |- ( ( C e. P. /\ A e. P. /\ x e. P. ) -> ( ( C +P. A ) +P. x ) = ( C +P. ( A +P. x ) ) )
15	3, 8, 11, 14	syl3anc 1249	. . . . 5 |- ( ( ( A <P B /\ C e. P. ) /\ ( x e. P. /\ ( A +P. x ) = B ) ) -> ( ( C +P. A ) +P. x ) = ( C +P. ( A +P. x ) ) )
16		oveq2 5927	. . . . . 6 |- ( ( A +P. x ) = B -> ( C +P. ( A +P. x ) ) = ( C +P. B ) )
17	16	ad2antll 491	. . . . 5 |- ( ( ( A <P B /\ C e. P. ) /\ ( x e. P. /\ ( A +P. x ) = B ) ) -> ( C +P. ( A +P. x ) ) = ( C +P. B ) )
18	15, 17	eqtrd 2226	. . . 4 |- ( ( ( A <P B /\ C e. P. ) /\ ( x e. P. /\ ( A +P. x ) = B ) ) -> ( ( C +P. A ) +P. x ) = ( C +P. B ) )
19	13, 18	breqtrd 4056	. . 3 |- ( ( ( A <P B /\ C e. P. ) /\ ( x e. P. /\ ( A +P. x ) = B ) ) -> ( C +P. A ) <P ( C +P. B ) )
20	2, 19	rexlimddv 2616	. 2 |- ( ( A <P B /\ C e. P. ) -> ( C +P. A ) <P ( C +P. B ) )
21	20	expcom 116	1 |- ( C e. P. -> ( A <P B -> ( C +P. A ) <P ( C +P. B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   E.wrex 2473   class class class wbr 4030  (class class class)co 5919   P.cnp 7353   +P. cpp 7355   <P cltp 7357

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 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

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 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-eprel 4321  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-1o 6471  df-2o 6472  df-oadd 6475  df-omul 6476  df-er 6589  df-ec 6591  df-qs 6595  df-ni 7366  df-pli 7367  df-mi 7368  df-lti 7369  df-plpq 7406  df-mpq 7407  df-enq 7409  df-nqqs 7410  df-plqqs 7411  df-mqqs 7412  df-1nqqs 7413  df-rq 7414  df-ltnqqs 7415  df-enq0 7486  df-nq0 7487  df-0nq0 7488  df-plq0 7489  df-mq0 7490  df-inp 7528  df-iplp 7530  df-iltp 7532

This theorem is referenced by:  ltaprg  7681