Theorem addcanpr 3946

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

Hypotheses

Ref	Expression
addcanpr.1	⊢ B ∈ V
addcanpr.2	⊢ C ∈ V

Assertion

Ref	Expression
addcanpr	⊢ ((A ∈ P ∧ B ∈ P) → ((A +P B) = (A +P C) → B = C))

Proof of Theorem addcanpr

Step	Hyp	Ref	Expression
1		eleq1 1149	. . . . . 6 ⊢ ((A +P B) = (A +P C) → ((A +P B) ∈ P ↔ (A +P C) ∈ P))
2		addcanpr.2	. . . . . . 7 ⊢ C ∈ V
3		dmplp 3909	. . . . . . 7 ⊢ dom +P = (P × P)
4		0npr 3890	. . . . . . 7 ⊢ ¬ ∅ ∈ P
5	2, 3, 4	ndmoprrcl 3060	. . . . . 6 ⊢ ((A +P C) ∈ P → (A ∈ P ∧ C ∈ P))
6	1, 5	syl6bi 187	. . . . 5 ⊢ ((A +P B) = (A +P C) → ((A +P B) ∈ P → (A ∈ P ∧ C ∈ P)))
7		addclpr 3914	. . . . 5 ⊢ ((A ∈ P ∧ B ∈ P) → (A +P B) ∈ P)
8	6, 7	syl5 22	. . . 4 ⊢ ((A +P B) = (A +P C) → ((A ∈ P ∧ B ∈ P) → (A ∈ P ∧ C ∈ P)))
9	8	com12 13	. . 3 ⊢ ((A ∈ P ∧ B ∈ P) → ((A +P B) = (A +P C) → (A ∈ P ∧ C ∈ P)))
10		addcanpr.1	. . . . . . . . . 10 ⊢ B ∈ V
11	10, 2	ltapr 3945	. . . . . . . . 9 ⊢ (A ∈ P → (B<P C ↔ (A +P B)<P (A +P C)))
12	2, 10	ltapr 3945	. . . . . . . . 9 ⊢ (A ∈ P → (C<P B ↔ (A +P C)<P (A +P B)))
13	11, 12	orbi12d 475	. . . . . . . 8 ⊢ (A ∈ P → ((B<P C ∨ C<P B) ↔ ((A +P B)<P (A +P C) ∨ (A +P C)<P (A +P B))))
14	13	negbid 463	. . . . . . 7 ⊢ (A ∈ P → (¬ (B<P C ∨ C<P B) ↔ ¬ ((A +P B)<P (A +P C) ∨ (A +P C)<P (A +P B))))
15	14	ad2antll 320	. . . . . 6 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (A ∈ P ∧ C ∈ P)) → (¬ (B<P C ∨ C<P B) ↔ ¬ ((A +P B)<P (A +P C) ∨ (A +P C)<P (A +P B))))
16		anandi 392	. . . . . . 7 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) ↔ ((A ∈ P ∧ B ∈ P) ∧ (A ∈ P ∧ C ∈ P)))
17		ltsopr 3930	. . . . . . . . 9 ⊢ <P Or P
18		sotrieq 2149	. . . . . . . . 9 ⊢ ((<P Or P ∧ (B ∈ P ∧ C ∈ P)) → (B = C ↔ ¬ (B<P C ∨ C<P B)))
19	17, 18	mpan 518	. . . . . . . 8 ⊢ ((B ∈ P ∧ C ∈ P) → (B = C ↔ ¬ (B<P C ∨ C<P B)))
20	19	adantl 305	. . . . . . 7 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → (B = C ↔ ¬ (B<P C ∨ C<P B)))
21	16, 20	sylbir 176	. . . . . 6 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (A ∈ P ∧ C ∈ P)) → (B = C ↔ ¬ (B<P C ∨ C<P B)))
22		sotrieq 2149	. . . . . . . 8 ⊢ ((<P Or P ∧ ((A +P B) ∈ P ∧ (A +P C) ∈ P)) → ((A +P B) = (A +P C) ↔ ¬ ((A +P B)<P (A +P C) ∨ (A +P C)<P (A +P B))))
23	17, 22	mpan 518	. . . . . . 7 ⊢ (((A +P B) ∈ P ∧ (A +P C) ∈ P) → ((A +P B) = (A +P C) ↔ ¬ ((A +P B)<P (A +P C) ∨ (A +P C)<P (A +P B))))
24		addclpr 3914	. . . . . . 7 ⊢ ((A ∈ P ∧ C ∈ P) → (A +P C) ∈ P)
25	23, 7, 24	syl2an 349	. . . . . 6 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (A ∈ P ∧ C ∈ P)) → ((A +P B) = (A +P C) ↔ ¬ ((A +P B)<P (A +P C) ∨ (A +P C)<P (A +P B))))
26	15, 21, 25	3bitr4d 424	. . . . 5 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (A ∈ P ∧ C ∈ P)) → (B = C ↔ (A +P B) = (A +P C)))
27	26	biimprd 136	. . . 4 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (A ∈ P ∧ C ∈ P)) → ((A +P B) = (A +P C) → B = C))
28	27	exp 291	. . 3 ⊢ ((A ∈ P ∧ B ∈ P) → ((A ∈ P ∧ C ∈ P) → ((A +P B) = (A +P C) → B = C)))
29	9, 28	syld 27	. 2 ⊢ ((A ∈ P ∧ B ∈ P) → ((A +P B) = (A +P C) → ((A +P B) = (A +P C) → B = C)))
30	29	pm2.43d 59	1 ⊢ ((A ∈ P ∧ B ∈ P) → ((A +P B) = (A +P C) → B = C))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348   class class class wbr 2054   Or wor 2059  (class class class)co 3001  Pcnp 3779   +_P cpp 3781  <_P cltp 3783

This theorem is referenced by:  enrer 3970  mulcmpblnr 3977  mulgt0sr 4008

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-plp 3882  df-ltp 3884

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