Theorem addcanpr 10969

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 10941	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) ∈ P)
2		eleq1 2824	. . . . 5 ⊢ ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → ((𝐴 +P 𝐵) ∈ P ↔ (𝐴 +P 𝐶) ∈ P))
3		dmplp 10935	. . . . . 6 ⊢ dom +P = (P × P)
4		0npr 10915	. . . . . 6 ⊢ ¬ ∅ ∈ P
5	3, 4	ndmovrcl 7553	. . . . 5 ⊢ ((𝐴 +P 𝐶) ∈ P → (𝐴 ∈ P ∧ 𝐶 ∈ P))
6	2, 5	biimtrdi 253	. . . 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 10968	. . . . . . . 8 ⊢ (𝐴 ∈ P → (𝐵<P 𝐶 ↔ (𝐴 +P 𝐵)<P (𝐴 +P 𝐶)))
9		ltapr 10968	. . . . . . . 8 ⊢ (𝐴 ∈ P → (𝐶<P 𝐵 ↔ (𝐴 +P 𝐶)<P (𝐴 +P 𝐵)))
10	8, 9	orbi12d 919	. . . . . . 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 318	. . . . . 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 727	. . . . 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 10955	. . . . . . 7 ⊢ <P Or P
14		sotrieq 5570	. . . . . . 7 ⊢ ((<P Or P ∧ (𝐵 ∈ P ∧ 𝐶 ∈ P)) → (𝐵 = 𝐶 ↔ ¬ (𝐵<P 𝐶 ∨ 𝐶<P 𝐵)))
15	13, 14	mpan 691	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 = 𝐶 ↔ ¬ (𝐵<P 𝐶 ∨ 𝐶<P 𝐵)))
16	15	ad2ant2l 747	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐴 ∈ P ∧ 𝐶 ∈ P)) → (𝐵 = 𝐶 ↔ ¬ (𝐵<P 𝐶 ∨ 𝐶<P 𝐵)))
17		addclpr 10941	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴 +P 𝐶) ∈ P)
18		sotrieq 5570	. . . . . . 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 691	. . . . . 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 597	. . . . 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 311	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐴 ∈ P ∧ 𝐶 ∈ P)) → (𝐵 = 𝐶 ↔ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)))
22	21	exbiri 811	. . 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 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   class class class wbr 5085   Or wor 5538  (class class class)co 7367  Pcnp 10782   +_P cpp 10784  <_P cltp 10786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-oadd 8409  df-omul 8410  df-er 8643  df-ni 10795  df-pli 10796  df-mi 10797  df-lti 10798  df-plpq 10831  df-mpq 10832  df-ltpq 10833  df-enq 10834  df-nq 10835  df-erq 10836  df-plq 10837  df-mq 10838  df-1nq 10839  df-rq 10840  df-ltnq 10841  df-np 10904  df-plp 10906  df-ltp 10908

This theorem is referenced by:  enrer  10986  mulcmpblnr  10994  mulgt0sr  11028