Theorem ltaprlem 7567

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 7562	. . . 4 |- ( A <P B -> E. x e. P. ( A +P. x ) = B )
2	1	adantr 274	. . 3 |- ( ( A <P B /\ C e. P. ) -> E. x e. P. ( A +P. x ) = B )
3		simplr 525	. . . . . 6 |- ( ( ( A <P B /\ C e. P. ) /\ ( x e. P. /\ ( A +P. x ) = B ) ) -> C e. P. )
4		ltrelpr 7454	. . . . . . . . . 10 |- <P C_ ( P. X. P. )
5	4	brel 4661	. . . . . . . . 9 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
6	5	simpld 111	. . . . . . . 8 |- ( A <P B -> A e. P. )
7	6	adantr 274	. . . . . . 7 |- ( ( A <P B /\ C e. P. ) -> A e. P. )
8	7	adantr 274	. . . . . 6 |- ( ( ( A <P B /\ C e. P. ) /\ ( x e. P. /\ ( A +P. x ) = B ) ) -> A e. P. )
9		addclpr 7486	. . . . . 6 |- ( ( C e. P. /\ A e. P. ) -> ( C +P. A ) e. P. )
10	3, 8, 9	syl2anc 409	. . . . 5 |- ( ( ( A <P B /\ C e. P. ) /\ ( x e. P. /\ ( A +P. x ) = B ) ) -> ( C +P. A ) e. P. )
11		simprl 526	. . . . 5 |- ( ( ( A <P B /\ C e. P. ) /\ ( x e. P. /\ ( A +P. x ) = B ) ) -> x e. P. )
12		ltaddpr 7546	. . . . 5 |- ( ( ( C +P. A ) e. P. /\ x e. P. ) -> ( C +P. A ) <P ( ( C +P. A ) +P. x ) )
13	10, 11, 12	syl2anc 409	. . . 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 7528	. . . . . 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 1233	. . . . 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 5858	. . . . . 6 |- ( ( A +P. x ) = B -> ( C +P. ( A +P. x ) ) = ( C +P. B ) )
17	16	ad2antll 488	. . . . 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 2203	. . . 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 4013	. . 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 2592	. 2 |- ( ( A <P B /\ C e. P. ) -> ( C +P. A ) <P ( C +P. B ) )
21	20	expcom 115	1 |- ( C e. P. -> ( A <P B -> ( C +P. A ) <P ( C +P. B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   E.wrex 2449   class class class wbr 3987  (class class class)co 5850   P.cnp 7240   +P. cpp 7242   <P cltp 7244

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-eprel 4272  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-irdg 6346  df-1o 6392  df-2o 6393  df-oadd 6396  df-omul 6397  df-er 6509  df-ec 6511  df-qs 6515  df-ni 7253  df-pli 7254  df-mi 7255  df-lti 7256  df-plpq 7293  df-mpq 7294  df-enq 7296  df-nqqs 7297  df-plqqs 7298  df-mqqs 7299  df-1nqqs 7300  df-rq 7301  df-ltnqqs 7302  df-enq0 7373  df-nq0 7374  df-0nq0 7375  df-plq0 7376  df-mq0 7377  df-inp 7415  df-iplp 7417  df-iltp 7419

This theorem is referenced by:  ltaprg  7568