Theorem addcanpr 10069

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 10041	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) ∈ P)
2		eleq1 2837	. . . . 5 ⊢ ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → ((𝐴 +P 𝐵) ∈ P ↔ (𝐴 +P 𝐶) ∈ P))
3		dmplp 10035	. . . . . 6 ⊢ dom +P = (P × P)
4		0npr 10015	. . . . . 6 ⊢ ¬ ∅ ∈ P
5	3, 4	ndmovrcl 6966	. . . . 5 ⊢ ((𝐴 +P 𝐶) ∈ P → (𝐴 ∈ P ∧ 𝐶 ∈ P))
6	2, 5	syl6bi 243	. . . 4 ⊢ ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → ((𝐴 +P 𝐵) ∈ P → (𝐴 ∈ P ∧ 𝐶 ∈ P)))
7	1, 6	syl5com 31	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → (𝐴 ∈ P ∧ 𝐶 ∈ P)))
8		ltapr 10068	. . . . . . . 8 ⊢ (𝐴 ∈ P → (𝐵<P 𝐶 ↔ (𝐴 +P 𝐵)<P (𝐴 +P 𝐶)))
9		ltapr 10068	. . . . . . . 8 ⊢ (𝐴 ∈ P → (𝐶<P 𝐵 ↔ (𝐴 +P 𝐶)<P (𝐴 +P 𝐵)))
10	8, 9	orbi12d 883	. . . . . . 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 307	. . . . . 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 697	. . . . 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 10055	. . . . . . 7 ⊢ <P Or P
14		sotrieq 5197	. . . . . . 7 ⊢ ((<P Or P ∧ (𝐵 ∈ P ∧ 𝐶 ∈ P)) → (𝐵 = 𝐶 ↔ ¬ (𝐵<P 𝐶 ∨ 𝐶<P 𝐵)))
15	13, 14	mpan 662	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 = 𝐶 ↔ ¬ (𝐵<P 𝐶 ∨ 𝐶<P 𝐵)))
16	15	ad2ant2l 732	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐴 ∈ P ∧ 𝐶 ∈ P)) → (𝐵 = 𝐶 ↔ ¬ (𝐵<P 𝐶 ∨ 𝐶<P 𝐵)))
17		addclpr 10041	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴 +P 𝐶) ∈ P)
18		sotrieq 5197	. . . . . . 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 662	. . . . . 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 575	. . . . 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 300	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐴 ∈ P ∧ 𝐶 ∈ P)) → (𝐵 = 𝐶 ↔ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)))
22	21	exbiri 808	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐴 ∈ P ∧ 𝐶 ∈ P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → 𝐵 = 𝐶)))
23	7, 22	syld 47	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → 𝐵 = 𝐶)))
24	23	pm2.43d 53	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → 𝐵 = 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382   ∨ wo 826   = wceq 1630   ∈ wcel 2144   class class class wbr 4784   Or wor 5169  (class class class)co 6792  Pcnp 9882   +_P cpp 9884  <_P cltp 9886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-omul 7717  df-er 7895  df-ni 9895  df-pli 9896  df-mi 9897  df-lti 9898  df-plpq 9931  df-mpq 9932  df-ltpq 9933  df-enq 9934  df-nq 9935  df-erq 9936  df-plq 9937  df-mq 9938  df-1nq 9939  df-rq 9940  df-ltnq 9941  df-np 10004  df-plp 10006  df-ltp 10008

This theorem is referenced by:  enrer  10087  mulcmpblnr  10093  mulgt0sr  10127