Theorem ltaprg 7420

Description: Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by Jim Kingdon, 26-Dec-2019.)

Assertion

Ref	Expression
ltaprg	|- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A <P B <-> ( C +P. A ) <P ( C +P. B ) ) )

Proof of Theorem ltaprg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltaprlem 7419	. . 3 |- ( C e. P. -> ( A <P B -> ( C +P. A ) <P ( C +P. B ) ) )
2	1	3ad2ant3 1004	. 2 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A <P B -> ( C +P. A ) <P ( C +P. B ) ) )
3		ltexpri 7414	. . . . 5 |- ( ( C +P. A ) <P ( C +P. B ) -> E. x e. P. ( ( C +P. A ) +P. x ) = ( C +P. B ) )
4	3	adantl 275	. . . 4 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C +P. A ) <P ( C +P. B ) ) -> E. x e. P. ( ( C +P. A ) +P. x ) = ( C +P. B ) )
5		simpl1 984	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> A e. P. )
6		simprl 520	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> x e. P. )
7		ltaddpr 7398	. . . . . . 7 |- ( ( A e. P. /\ x e. P. ) -> A <P ( A +P. x ) )
8	5, 6, 7	syl2anc 408	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> A <P ( A +P. x ) )
9		addassprg 7380	. . . . . . . . . . . 12 |- ( ( C e. P. /\ A e. P. /\ x e. P. ) -> ( ( C +P. A ) +P. x ) = ( C +P. ( A +P. x ) ) )
10	9	3com12 1185	. . . . . . . . . . 11 |- ( ( A e. P. /\ C e. P. /\ x e. P. ) -> ( ( C +P. A ) +P. x ) = ( C +P. ( A +P. x ) ) )
11	10	3expa 1181	. . . . . . . . . 10 |- ( ( ( A e. P. /\ C e. P. ) /\ x e. P. ) -> ( ( C +P. A ) +P. x ) = ( C +P. ( A +P. x ) ) )
12	11	adantrr 470	. . . . . . . . 9 |- ( ( ( A e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> ( ( C +P. A ) +P. x ) = ( C +P. ( A +P. x ) ) )
13		simprr 521	. . . . . . . . 9 |- ( ( ( A e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> ( ( C +P. A ) +P. x ) = ( C +P. B ) )
14	12, 13	eqtr3d 2172	. . . . . . . 8 |- ( ( ( A e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> ( C +P. ( A +P. x ) ) = ( C +P. B ) )
15	14	3adantl2 1138	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> ( C +P. ( A +P. x ) ) = ( C +P. B ) )
16		simpl3 986	. . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> C e. P. )
17		addclpr 7338	. . . . . . . . 9 |- ( ( A e. P. /\ x e. P. ) -> ( A +P. x ) e. P. )
18	5, 6, 17	syl2anc 408	. . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> ( A +P. x ) e. P. )
19		simpl2 985	. . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> B e. P. )
20		addcanprg 7417	. . . . . . . 8 |- ( ( C e. P. /\ ( A +P. x ) e. P. /\ B e. P. ) -> ( ( C +P. ( A +P. x ) ) = ( C +P. B ) -> ( A +P. x ) = B ) )
21	16, 18, 19, 20	syl3anc 1216	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> ( ( C +P. ( A +P. x ) ) = ( C +P. B ) -> ( A +P. x ) = B ) )
22	15, 21	mpd 13	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> ( A +P. x ) = B )
23	8, 22	breqtrd 3949	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> A <P B )
24	23	adantlr 468	. . . 4 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C +P. A ) <P ( C +P. B ) ) /\ ( x e. P. /\ ( ( C +P. A ) +P. x ) = ( C +P. B ) ) ) -> A <P B )
25	4, 24	rexlimddv 2552	. . 3 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( C +P. A ) <P ( C +P. B ) ) -> A <P B )
26	25	ex 114	. 2 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( C +P. A ) <P ( C +P. B ) -> A <P B ) )
27	2, 26	impbid 128	1 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A <P B <-> ( C +P. A ) <P ( C +P. B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   E.wrex 2415   class class class wbr 3924  (class class class)co 5767   P.cnp 7092   +P. cpp 7094   <P cltp 7096

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-2o 6307  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-enq0 7225  df-nq0 7226  df-0nq0 7227  df-plq0 7228  df-mq0 7229  df-inp 7267  df-iplp 7269  df-iltp 7271

This theorem is referenced by:  prplnqu  7421  addextpr  7422  caucvgprlemcanl  7445  caucvgprprlemnkltj  7490  caucvgprprlemnbj  7494  caucvgprprlemmu  7496  caucvgprprlemloc  7504  caucvgprprlemexbt  7507  caucvgprprlemexb  7508  caucvgprprlemaddq  7509  caucvgprprlem1  7510  caucvgprprlem2  7511  ltsrprg  7548  gt0srpr  7549  lttrsr  7563  ltsosr  7565  ltasrg  7571  prsrlt  7588  ltpsrprg  7604  map2psrprg  7606