Theorem lteupri 7934

Description: The difference from ltexpri 7930 is unique. (Contributed by Jim Kingdon, 7-Jul-2021.)

Assertion

Ref	Expression
lteupri	|- ( A <P B -> E! x e. P. ( A +P. x ) = B )

Distinct variable groups:   x, A   x, B

Proof of Theorem lteupri

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltexpri 7930	. 2 |- ( A <P B -> E. x e. P. ( A +P. x ) = B )
2		ltrelpr 7822	. . . . 5 |- <P C_ ( P. X. P. )
3	2	brel 4804	. . . 4 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
4	3	simpld 112	. . 3 |- ( A <P B -> A e. P. )
5		eqtr3 2254	. . . . . . . 8 |- ( ( ( A +P. x ) = B /\ ( A +P. y ) = B ) -> ( A +P. x ) = ( A +P. y ) )
6		addcanprg 7933	. . . . . . . 8 |- ( ( A e. P. /\ x e. P. /\ y e. P. ) -> ( ( A +P. x ) = ( A +P. y ) -> x = y ) )
7	5, 6	syl5 32	. . . . . . 7 |- ( ( A e. P. /\ x e. P. /\ y e. P. ) -> ( ( ( A +P. x ) = B /\ ( A +P. y ) = B ) -> x = y ) )
8	7	3expa 1230	. . . . . 6 |- ( ( ( A e. P. /\ x e. P. ) /\ y e. P. ) -> ( ( ( A +P. x ) = B /\ ( A +P. y ) = B ) -> x = y ) )
9	8	ralrimiva 2617	. . . . 5 |- ( ( A e. P. /\ x e. P. ) -> A. y e. P. ( ( ( A +P. x ) = B /\ ( A +P. y ) = B ) -> x = y ) )
10	9	ralrimiva 2617	. . . 4 |- ( A e. P. -> A. x e. P. A. y e. P. ( ( ( A +P. x ) = B /\ ( A +P. y ) = B ) -> x = y ) )
11		oveq2 6060	. . . . . 6 |- ( x = y -> ( A +P. x ) = ( A +P. y ) )
12	11	eqeq1d 2243	. . . . 5 |- ( x = y -> ( ( A +P. x ) = B <-> ( A +P. y ) = B ) )
13	12	rmo4 3012	. . . 4 |- ( E* x e. P. ( A +P. x ) = B <-> A. x e. P. A. y e. P. ( ( ( A +P. x ) = B /\ ( A +P. y ) = B ) -> x = y ) )
14	10, 13	sylibr 134	. . 3 |- ( A e. P. -> E* x e. P. ( A +P. x ) = B )
15	4, 14	syl 14	. 2 |- ( A <P B -> E* x e. P. ( A +P. x ) = B )
16		reu5 2764	. 2 |- ( E! x e. P. ( A +P. x ) = B <-> ( E. x e. P. ( A +P. x ) = B /\ E* x e. P. ( A +P. x ) = B ) )
17	1, 15, 16	sylanbrc 417	1 |- ( A <P B -> E! x e. P. ( A +P. x ) = B )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   E!wreu 2524   E*wrmo 2525   class class class wbr 4111  (class class class)co 6052   P.cnp 7608   +P. cpp 7610   <P cltp 7612

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 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712

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-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-eprel 4412  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-1o 6649  df-2o 6650  df-oadd 6653  df-omul 6654  df-er 6769  df-ec 6771  df-qs 6775  df-ni 7621  df-pli 7622  df-mi 7623  df-lti 7624  df-plpq 7661  df-mpq 7662  df-enq 7664  df-nqqs 7665  df-plqqs 7666  df-mqqs 7667  df-1nqqs 7668  df-rq 7669  df-ltnqqs 7670  df-enq0 7741  df-nq0 7742  df-0nq0 7743  df-plq0 7744  df-mq0 7745  df-inp 7783  df-iplp 7785  df-iltp 7787

This theorem is referenced by:  srpospr  8100