Theorem addcanpr 11019

Description: Addition cancellation law for positive reals. Proposition 9-3.5(vi) of [Gleason] p. 123. (Contributed by NM, 9-Apr-1996.) (New usage is discouraged.)

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		addclpr 10991	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) ∈ P)
2		eleq1 2853	. . . . 5 ⊢ ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → ((𝐴 +P 𝐵) ∈ P ↔ (𝐴 +P 𝐶) ∈ P))
3		dmplp 10985	. . . . . 6 ⊢ dom +P = (P × P)
4		0npr 10965	. . . . . 6 ⊢ ¬ ∅ ∈ P
5	3, 4	ndmovrcl 7586	. . . . 5 ⊢ ((𝐴 +P 𝐶) ∈ P → (𝐴 ∈ P ∧ 𝐶 ∈ P))
6	2, 5	biimtrdi 256	. . . 4 ⊢ ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → ((𝐴 +P 𝐵) ∈ P → (𝐴 ∈ P ∧ 𝐶 ∈ P)))
7	1, 6	syl5com 32	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → (𝐴 ∈ P ∧ 𝐶 ∈ P)))
8		ltapr 11018	. . . . . . . 8 ⊢ (𝐴 ∈ P → (𝐵<P 𝐶 ↔ (𝐴 +P 𝐵)<P (𝐴 +P 𝐶)))
9		ltapr 11018	. . . . . . . 8 ⊢ (𝐴 ∈ P → (𝐶<P 𝐵 ↔ (𝐴 +P 𝐶)<P (𝐴 +P 𝐵)))
10	8, 9	orbi12d 931	. . . . . . 7 ⊢ (𝐴 ∈ P → ((𝐵<P 𝐶 ∨ 𝐶<P 𝐵) ↔ ((𝐴 +P 𝐵)<P (𝐴 +P 𝐶) ∨ (𝐴 +P 𝐶)<P (𝐴 +P 𝐵))))
11	10	notbid 321	. . . . . 6 ⊢ (𝐴 ∈ P → (¬ (𝐵<P 𝐶 ∨ 𝐶<P 𝐵) ↔ ¬ ((𝐴 +P 𝐵)<P (𝐴 +P 𝐶) ∨ (𝐴 +P 𝐶)<P (𝐴 +P 𝐵))))
12	11	ad2antrr 738	. . . . 5 ⊢ (((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))))
13		ltsopr 11005	. . . . . . 7 ⊢ <P Or P
14		sotrieq 5590	. . . . . . 7 ⊢ ((<P Or P ∧ (𝐵 ∈ P ∧ 𝐶 ∈ P)) → (𝐵 = 𝐶 ↔ ¬ (𝐵<P 𝐶 ∨ 𝐶<P 𝐵)))
15	13, 14	mpan 702	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 = 𝐶 ↔ ¬ (𝐵<P 𝐶 ∨ 𝐶<P 𝐵)))
16	15	ad2ant2l 758	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐴 ∈ P ∧ 𝐶 ∈ P)) → (𝐵 = 𝐶 ↔ ¬ (𝐵<P 𝐶 ∨ 𝐶<P 𝐵)))
17		addclpr 10991	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴 +P 𝐶) ∈ P)
18		sotrieq 5590	. . . . . . 7 ⊢ ((<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))))
19	13, 18	mpan 702	. . . . . 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))))
20	1, 17, 19	syl2an 607	. . . . 5 ⊢ (((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))))
21	12, 16, 20	3bitr4d 314	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐴 ∈ P ∧ 𝐶 ∈ P)) → (𝐵 = 𝐶 ↔ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)))
22	21	exbiri 822	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐴 ∈ P ∧ 𝐶 ∈ P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → 𝐵 = 𝐶)))
23	7, 22	syld 48	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → 𝐵 = 𝐶)))
24	23	pm2.43d 54	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → 𝐵 = 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1563   ∈ wcel 2145   class class class wbr 5104   Or wor 5558  (class class class)co 7400  Pcnp 10832   +_P cpp 10834  <_P cltp 10836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722  ax-inf2 9598

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4908  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6291  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-oadd 8445  df-omul 8446  df-er 8682  df-ni 10845  df-pli 10846  df-mi 10847  df-lti 10848  df-plpq 10881  df-mpq 10882  df-ltpq 10883  df-enq 10884  df-nq 10885  df-erq 10886  df-plq 10887  df-mq 10888  df-1nq 10889  df-rq 10890  df-ltnq 10891  df-np 10954  df-plp 10956  df-ltp 10958

This theorem is referenced by:  enrer  11036  mulcmpblnr  11044  mulgt0sr  11078