Theorem ltaprlem 7949

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 7944	. . . 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 529	. . . . . 6 |- ( ( ( A <P B /\ C e. P. ) /\ ( x e. P. /\ ( A +P. x ) = B ) ) -> C e. P. )
4		ltrelpr 7836	. . . . . . . . . 10 |- <P C_ ( P. X. P. )
5	4	brel 4807	. . . . . . . . 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 7868	. . . . . 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 531	. . . . 5 |- ( ( ( A <P B /\ C e. P. ) /\ ( x e. P. /\ ( A +P. x ) = B ) ) -> x e. P. )
12		ltaddpr 7928	. . . . 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 7910	. . . . . 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 1274	. . . . 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 6066	. . . . . 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 2267	. . . 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 4140	. . 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 2667	. 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 1398   e. wcel 2205   E.wrex 2523   class class class wbr 4114  (class class class)co 6058   P.cnp 7622   +P. cpp 7624   <P cltp 7626

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-2o 6661  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684  df-enq0 7755  df-nq0 7756  df-0nq0 7757  df-plq0 7758  df-mq0 7759  df-inp 7797  df-iplp 7799  df-iltp 7801

This theorem is referenced by:  ltaprg  7950