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Mirrors > Home > MPE Home > Th. List > addcanpr | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
addcanpr | ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → 𝐵 = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addclpr 10440 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) ∈ P) | |
2 | eleq1 2900 | . . . . 5 ⊢ ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → ((𝐴 +P 𝐵) ∈ P ↔ (𝐴 +P 𝐶) ∈ P)) | |
3 | dmplp 10434 | . . . . . 6 ⊢ dom +P = (P × P) | |
4 | 0npr 10414 | . . . . . 6 ⊢ ¬ ∅ ∈ P | |
5 | 3, 4 | ndmovrcl 7334 | . . . . 5 ⊢ ((𝐴 +P 𝐶) ∈ P → (𝐴 ∈ P ∧ 𝐶 ∈ P)) |
6 | 2, 5 | syl6bi 255 | . . . 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 10467 | . . . . . . . 8 ⊢ (𝐴 ∈ P → (𝐵<P 𝐶 ↔ (𝐴 +P 𝐵)<P (𝐴 +P 𝐶))) | |
9 | ltapr 10467 | . . . . . . . 8 ⊢ (𝐴 ∈ P → (𝐶<P 𝐵 ↔ (𝐴 +P 𝐶)<P (𝐴 +P 𝐵))) | |
10 | 8, 9 | orbi12d 915 | . . . . . . 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 320 | . . . . . 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 724 | . . . . 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 10454 | . . . . . . 7 ⊢ <P Or P | |
14 | sotrieq 5502 | . . . . . . 7 ⊢ ((<P Or P ∧ (𝐵 ∈ P ∧ 𝐶 ∈ P)) → (𝐵 = 𝐶 ↔ ¬ (𝐵<P 𝐶 ∨ 𝐶<P 𝐵))) | |
15 | 13, 14 | mpan 688 | . . . . . 6 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 = 𝐶 ↔ ¬ (𝐵<P 𝐶 ∨ 𝐶<P 𝐵))) |
16 | 15 | ad2ant2l 744 | . . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐴 ∈ P ∧ 𝐶 ∈ P)) → (𝐵 = 𝐶 ↔ ¬ (𝐵<P 𝐶 ∨ 𝐶<P 𝐵))) |
17 | addclpr 10440 | . . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴 +P 𝐶) ∈ P) | |
18 | sotrieq 5502 | . . . . . . 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 688 | . . . . . 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 313 | . . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐴 ∈ P ∧ 𝐶 ∈ P)) → (𝐵 = 𝐶 ↔ (𝐴 +P 𝐵) = (𝐴 +P 𝐶))) |
22 | 21 | exbiri 809 | . . 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)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 Or wor 5473 (class class class)co 7156 Pcnp 10281 +P cpp 10283 <P cltp 10285 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-omul 8107 df-er 8289 df-ni 10294 df-pli 10295 df-mi 10296 df-lti 10297 df-plpq 10330 df-mpq 10331 df-ltpq 10332 df-enq 10333 df-nq 10334 df-erq 10335 df-plq 10336 df-mq 10337 df-1nq 10338 df-rq 10339 df-ltnq 10340 df-np 10403 df-plp 10405 df-ltp 10407 |
This theorem is referenced by: enrer 10485 mulcmpblnr 10493 mulgt0sr 10527 |
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