Theorem ltaprlem 7766

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 7761	. . . 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 7653	. . . . . . . . . 10 |- <P C_ ( P. X. P. )
5	4	brel 4745	. . . . . . . . 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 7685	. . . . . 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 7745	. . . . 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 7727	. . . . . 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 1250	. . . . 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 5975	. . . . . 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 2240	. . . 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 4085	. . 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 2630	. 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 1373   e. wcel 2178   E.wrex 2487   class class class wbr 4059  (class class class)co 5967   P.cnp 7439   +P. cpp 7441   <P cltp 7443

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

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 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-eprel 4354  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-1o 6525  df-2o 6526  df-oadd 6529  df-omul 6530  df-er 6643  df-ec 6645  df-qs 6649  df-ni 7452  df-pli 7453  df-mi 7454  df-lti 7455  df-plpq 7492  df-mpq 7493  df-enq 7495  df-nqqs 7496  df-plqqs 7497  df-mqqs 7498  df-1nqqs 7499  df-rq 7500  df-ltnqqs 7501  df-enq0 7572  df-nq0 7573  df-0nq0 7574  df-plq0 7575  df-mq0 7576  df-inp 7614  df-iplp 7616  df-iltp 7618

This theorem is referenced by:  ltaprg  7767