Theorem ltaprlem 6302

Description: Lemma for Proposition 9-3.5(v) of [Gleason] p. 123.

Hypotheses

Ref	Expression
ltaprlem.1	|- A e. _V
ltaprlem.2	|- B e. _V

Assertion

Ref	Expression
ltaprlem	|- (C e. P. -> (A <P B -> (C +P. A) <P (C +P. B)))

Proof of Theorem ltaprlem

Step	Hyp	Ref	Expression
1		ltaprlem.2	. . . . . 6 |- B e. _V
2	1	ltexpri 6301	. . . . 5 |- (A <P B -> E.x(x e. P. /\ (A +P. x) = B))
3		opreq2 4890	. . . . . . . . . . . . 13 |- ((A +P. x) = B -> (C +P. (A +P. x)) = (C +P. B))
4		ltaprlem.1	. . . . . . . . . . . . . 14 |- A e. _V
5		visset 2295	. . . . . . . . . . . . . 14 |- x e. _V
6	4, 5	addasspr 6276	. . . . . . . . . . . . 13 |- ((C +P. A) +P. x) = (C +P. (A +P. x))
7	3, 6	syl5eq 1940	. . . . . . . . . . . 12 |- ((A +P. x) = B -> ((C +P. A) +P. x) = (C +P. B))
8	7	breq2d 3350	. . . . . . . . . . 11 |- ((A +P. x) = B -> ((C +P. A) <P ((C +P. A) +P. x) <-> (C +P. A) <P (C +P. B)))
9		ltaddpr 6292	. . . . . . . . . . 11 |- (((C +P. A) e. P. /\ x e. P.) -> (C +P. A) <P ((C +P. A) +P. x))
10	8, 9	syl5bi 225	. . . . . . . . . 10 |- ((A +P. x) = B -> (((C +P. A) e. P. /\ x e. P.) -> (C +P. A) <P (C +P. B)))
11	10	exp3a 405	. . . . . . . . 9 |- ((A +P. x) = B -> ((C +P. A) e. P. -> (x e. P. -> (C +P. A) <P (C +P. B))))
12		addclpr 6272	. . . . . . . . 9 |- ((C e. P. /\ A e. P.) -> (C +P. A) e. P.)
13	11, 12	syl5 20	. . . . . . . 8 |- ((A +P. x) = B -> ((C e. P. /\ A e. P.) -> (x e. P. -> (C +P. A) <P (C +P. B))))
14	13	com3r 39	. . . . . . 7 |- (x e. P. -> ((A +P. x) = B -> ((C e. P. /\ A e. P.) -> (C +P. A) <P (C +P. B))))
15	14	imp 377	. . . . . 6 |- ((x e. P. /\ (A +P. x) = B) -> ((C e. P. /\ A e. P.) -> (C +P. A) <P (C +P. B)))
16	15	19.23aiv 1674	. . . . 5 |- (E.x(x e. P. /\ (A +P. x) = B) -> ((C e. P. /\ A e. P.) -> (C +P. A) <P (C +P. B)))
17	2, 16	syl 12	. . . 4 |- (A <P B -> ((C e. P. /\ A e. P.) -> (C +P. A) <P (C +P. B)))
18		ltrelpr 6253	. . . . . 6 |- <P C_ (P. X. P.)
19	1, 18	brel 4048	. . . . 5 |- (A <P B -> (A e. P. /\ B e. P.))
20	19	simplld 348	. . . 4 |- (A <P B -> A e. P.)
21	17, 20	sylan2i 514	. . 3 |- (A <P B -> ((C e. P. /\ A <P B) -> (C +P. A) <P (C +P. B)))
22	21	exp3a 405	. 2 |- (A <P B -> (C e. P. -> (A <P B -> (C +P. A) <P (C +P. B))))
23	22	pm2.43b 81	1 |- (C e. P. -> (A <P B -> (C +P. A) <P (C +P. B)))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292   class class class wbr 3338  (class class class)co 4884  P.cnp 6137   +P. cpp 6139   <P cltp 6141

This theorem is referenced by:  ltapr 6303

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-plp 6240  df-ltp 6242

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