MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  addnidpi Structured version   Visualization version   GIF version

Theorem addnidpi 9683
Description: There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.) (New usage is discouraged.)
Assertion
Ref Expression
addnidpi (𝐴N → ¬ (𝐴 +N 𝐵) = 𝐴)

Proof of Theorem addnidpi
StepHypRef Expression
1 pinn 9660 . . . . 5 (𝐴N𝐴 ∈ ω)
2 elni2 9659 . . . . . 6 (𝐵N ↔ (𝐵 ∈ ω ∧ ∅ ∈ 𝐵))
3 nnaordi 7658 . . . . . . . . 9 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝐵 → (𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝐵)))
4 nna0 7644 . . . . . . . . . . . 12 (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)
54eleq1d 2683 . . . . . . . . . . 11 (𝐴 ∈ ω → ((𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝐵) ↔ 𝐴 ∈ (𝐴 +𝑜 𝐵)))
6 nnord 7035 . . . . . . . . . . . . . 14 (𝐴 ∈ ω → Ord 𝐴)
7 ordirr 5710 . . . . . . . . . . . . . 14 (Ord 𝐴 → ¬ 𝐴𝐴)
86, 7syl 17 . . . . . . . . . . . . 13 (𝐴 ∈ ω → ¬ 𝐴𝐴)
9 eleq2 2687 . . . . . . . . . . . . . 14 ((𝐴 +𝑜 𝐵) = 𝐴 → (𝐴 ∈ (𝐴 +𝑜 𝐵) ↔ 𝐴𝐴))
109notbid 308 . . . . . . . . . . . . 13 ((𝐴 +𝑜 𝐵) = 𝐴 → (¬ 𝐴 ∈ (𝐴 +𝑜 𝐵) ↔ ¬ 𝐴𝐴))
118, 10syl5ibrcom 237 . . . . . . . . . . . 12 (𝐴 ∈ ω → ((𝐴 +𝑜 𝐵) = 𝐴 → ¬ 𝐴 ∈ (𝐴 +𝑜 𝐵)))
1211con2d 129 . . . . . . . . . . 11 (𝐴 ∈ ω → (𝐴 ∈ (𝐴 +𝑜 𝐵) → ¬ (𝐴 +𝑜 𝐵) = 𝐴))
135, 12sylbid 230 . . . . . . . . . 10 (𝐴 ∈ ω → ((𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝐵) → ¬ (𝐴 +𝑜 𝐵) = 𝐴))
1413adantl 482 . . . . . . . . 9 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → ((𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝐵) → ¬ (𝐴 +𝑜 𝐵) = 𝐴))
153, 14syld 47 . . . . . . . 8 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝐵 → ¬ (𝐴 +𝑜 𝐵) = 𝐴))
1615expcom 451 . . . . . . 7 (𝐴 ∈ ω → (𝐵 ∈ ω → (∅ ∈ 𝐵 → ¬ (𝐴 +𝑜 𝐵) = 𝐴)))
1716imp32 449 . . . . . 6 ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ ∅ ∈ 𝐵)) → ¬ (𝐴 +𝑜 𝐵) = 𝐴)
182, 17sylan2b 492 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵N) → ¬ (𝐴 +𝑜 𝐵) = 𝐴)
191, 18sylan 488 . . . 4 ((𝐴N𝐵N) → ¬ (𝐴 +𝑜 𝐵) = 𝐴)
20 addpiord 9666 . . . . 5 ((𝐴N𝐵N) → (𝐴 +N 𝐵) = (𝐴 +𝑜 𝐵))
2120eqeq1d 2623 . . . 4 ((𝐴N𝐵N) → ((𝐴 +N 𝐵) = 𝐴 ↔ (𝐴 +𝑜 𝐵) = 𝐴))
2219, 21mtbird 315 . . 3 ((𝐴N𝐵N) → ¬ (𝐴 +N 𝐵) = 𝐴)
2322a1d 25 . 2 ((𝐴N𝐵N) → (𝐴N → ¬ (𝐴 +N 𝐵) = 𝐴))
24 dmaddpi 9672 . . . . . 6 dom +N = (N × N)
2524ndmov 6783 . . . . 5 (¬ (𝐴N𝐵N) → (𝐴 +N 𝐵) = ∅)
2625eqeq1d 2623 . . . 4 (¬ (𝐴N𝐵N) → ((𝐴 +N 𝐵) = 𝐴 ↔ ∅ = 𝐴))
27 0npi 9664 . . . . 5 ¬ ∅ ∈ N
28 eleq1 2686 . . . . 5 (∅ = 𝐴 → (∅ ∈ N𝐴N))
2927, 28mtbii 316 . . . 4 (∅ = 𝐴 → ¬ 𝐴N)
3026, 29syl6bi 243 . . 3 (¬ (𝐴N𝐵N) → ((𝐴 +N 𝐵) = 𝐴 → ¬ 𝐴N))
3130con2d 129 . 2 (¬ (𝐴N𝐵N) → (𝐴N → ¬ (𝐴 +N 𝐵) = 𝐴))
3223, 31pm2.61i 176 1 (𝐴N → ¬ (𝐴 +N 𝐵) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  c0 3897  Ord word 5691  (class class class)co 6615  ωcom 7027   +𝑜 coa 7517  Ncnpi 9626   +N cpli 9627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-oadd 7524  df-ni 9654  df-pli 9655
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator