Theorem addcanpr 5139

Description: Addition cancellation law for positive reals. Proposition 9-3.5(vi) of [Gleason] p. 123.

Hypotheses

Ref	Expression
addcanpr.1	|- B e. V
addcanpr.2	|- C e. V

Assertion

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

Proof of Theorem addcanpr

Step	Hyp	Ref	Expression
1		eleq1 1533	. . . . 5 |- ((A +P. B) = (A +P. C) -> ((A +P. B) e. P. <-> (A +P. C) e. P.))
2		addcanpr.2	. . . . . 6 |- C e. V
3		dmplp 5102	. . . . . 6 |- dom +P. = (P. X. P.)
4		0npr 5083	. . . . . 6 |- -. (/) e. P.
5	2, 3, 4	ndmoprrcl 4043	. . . . 5 |- ((A +P. C) e. P. -> (A e. P. /\ C e. P.))
6	1, 5	syl6bi 214	. . . 4 |- ((A +P. B) = (A +P. C) -> ((A +P. B) e. P. -> (A e. P. /\ C e. P.)))
7		addclpr 5107	. . . 4 |- ((A e. P. /\ B e. P.) -> (A +P. B) e. P.)
8	6, 7	syl5com 52	. . 3 |- ((A e. P. /\ B e. P.) -> ((A +P. B) = (A +P. C) -> (A e. P. /\ C e. P.)))
9		addcanpr.1	. . . . . . . . . 10 |- B e. V
10	9, 2	ltapr 5138	. . . . . . . . 9 |- (A e. P. -> (B <P C <-> (A +P. B) <P (A +P. C)))
11	2, 9	ltapr 5138	. . . . . . . . 9 |- (A e. P. -> (C <P B <-> (A +P. C) <P (A +P. B)))
12	10, 11	orbi12d 626	. . . . . . . 8 |- (A e. P. -> ((B <P C \/ C <P B) <-> ((A +P. B) <P (A +P. C) \/ (A +P. C) <P (A +P. B))))
13	12	negbid 610	. . . . . . 7 |- (A e. P. -> (-. (B <P C \/ C <P B) <-> -. ((A +P. B) <P (A +P. C) \/ (A +P. C) <P (A +P. B))))
14	13	ad2antrr 404	. . . . . 6 |- (((A e. P. /\ B e. P.) /\ (A e. P. /\ C e. P.)) -> (-. (B <P C \/ C <P B) <-> -. ((A +P. B) <P (A +P. C) \/ (A +P. C) <P (A +P. B))))
15		ltsopr 5123	. . . . . . . 8 |- <P Or P.
16		sotrieq 2858	. . . . . . . 8 |- (( <P Or P. /\ (B e. P. /\ C e. P.)) -> (B = C <-> -. (B <P C \/ C <P B)))
17	15, 16	mpan 694	. . . . . . 7 |- ((B e. P. /\ C e. P.) -> (B = C <-> -. (B <P C \/ C <P B)))
18	17	ad2ant2l 408	. . . . . 6 |- (((A e. P. /\ B e. P.) /\ (A e. P. /\ C e. P.)) -> (B = C <-> -. (B <P C \/ C <P B)))
19		sotrieq 2858	. . . . . . . 8 |- (( <P Or P. /\ ((A +P. B) e. P. /\ (A +P. C) e. P.)) -> ((A +P. B) = (A +P. C) <-> -. ((A +P. B) <P (A +P. C) \/ (A +P. C) <P (A +P. B))))
20	15, 19	mpan 694	. . . . . . 7 |- (((A +P. B) e. P. /\ (A +P. C) e. P.) -> ((A +P. B) = (A +P. C) <-> -. ((A +P. B) <P (A +P. C) \/ (A +P. C) <P (A +P. B))))
21		addclpr 5107	. . . . . . 7 |- ((A e. P. /\ C e. P.) -> (A +P. C) e. P.)
22	20, 7, 21	syl2an 454	. . . . . 6 |- (((A e. P. /\ B e. P.) /\ (A e. P. /\ C e. P.)) -> ((A +P. B) = (A +P. C) <-> -. ((A +P. B) <P (A +P. C) \/ (A +P. C) <P (A +P. B))))
23	14, 18, 22	3bitr4d 549	. . . . 5 |- (((A e. P. /\ B e. P.) /\ (A e. P. /\ C e. P.)) -> (B = C <-> (A +P. B) = (A +P. C)))
24	23	biimprd 154	. . . 4 |- (((A e. P. /\ B e. P.) /\ (A e. P. /\ C e. P.)) -> ((A +P. B) = (A +P. C) -> B = C))
25	24	ex 373	. . 3 |- ((A e. P. /\ B e. P.) -> ((A e. P. /\ C e. P.) -> ((A +P. B) = (A +P. C) -> B = C)))
26	8, 25	syld 27	. 2 |- ((A e. P. /\ B e. P.) -> ((A +P. B) = (A +P. C) -> ((A +P. B) = (A +P. C) -> B = C)))
27	26	pm2.43d 65	1 |- ((A e. P. /\ B e. P.) -> ((A +P. B) = (A +P. C) -> B = C))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 955   e. wcel 957  Vcvv 1809   class class class wbr 2616   Or wor 2836  (class class class)co 3960  P.cnp 4972   +P. cpp 4974   <P cltp 4976

This theorem is referenced by:  enrer 5163  mulcmpblnr 5170  mulgt0sr 5201

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-nul 2707  ax-pow 2739  ax-pr 2776  ax-un 2863  ax-inf2 4612

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-reu 1650  df-rab 1651  df-v 1810  df-sbc 1940  df-csb 2000  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-pss 2053  df-nul 2279  df-if 2360  df-pw 2400  df-sn 2410  df-pr 2411  df-tp 2413  df-op 2414  df-uni 2501  df-int 2531  df-iun 2565  df-br 2617  df-opab 2664  df-tr 2678  df-eprel 2829  df-id 2832  df-po 2837  df-so 2847  df-fr 2914  df-we 2931  df-ord 2948  df-on 2949  df-lim 2950  df-suc 2951  df-om 3129  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-fv 3195  df-rdg 3929  df-opr 3962  df-oprab 3963  df-1st 4076  df-2nd 4077  df-1o 4130  df-oadd 4132  df-omul 4133  df-er 4258  df-ec 4260  df-qs 4263  df-ni 4987  df-pli 4988  df-mi 4989  df-lti 4990  df-plpq 5022  df-mpq 5023  df-enq 5024  df-nq 5025  df-plq 5026  df-mq 5027  df-rq 5028  df-ltq 5029  df-1q 5030  df-np 5073  df-plp 5075  df-ltp 5077

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