MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  o2p2e4 Structured version   Visualization version   GIF version

Theorem o2p2e4 7666
Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 5767. For the usual proof using complex numbers, see 2p2e4 11182. (Contributed by NM, 18-Aug-2021.)
Assertion
Ref Expression
o2p2e4 (2𝑜 +𝑜 2𝑜) = 4𝑜

Proof of Theorem o2p2e4
StepHypRef Expression
1 2on 7613 . . . 4 2𝑜 ∈ On
2 1on 7612 . . . 4 1𝑜 ∈ On
3 oasuc 7649 . . . 4 ((2𝑜 ∈ On ∧ 1𝑜 ∈ On) → (2𝑜 +𝑜 suc 1𝑜) = suc (2𝑜 +𝑜 1𝑜))
41, 2, 3mp2an 708 . . 3 (2𝑜 +𝑜 suc 1𝑜) = suc (2𝑜 +𝑜 1𝑜)
5 df-2o 7606 . . . 4 2𝑜 = suc 1𝑜
65oveq2i 6701 . . 3 (2𝑜 +𝑜 2𝑜) = (2𝑜 +𝑜 suc 1𝑜)
7 df-3o 7607 . . . . 5 3𝑜 = suc 2𝑜
8 oa1suc 7656 . . . . . 6 (2𝑜 ∈ On → (2𝑜 +𝑜 1𝑜) = suc 2𝑜)
91, 8ax-mp 5 . . . . 5 (2𝑜 +𝑜 1𝑜) = suc 2𝑜
107, 9eqtr4i 2676 . . . 4 3𝑜 = (2𝑜 +𝑜 1𝑜)
11 suceq 5828 . . . 4 (3𝑜 = (2𝑜 +𝑜 1𝑜) → suc 3𝑜 = suc (2𝑜 +𝑜 1𝑜))
1210, 11ax-mp 5 . . 3 suc 3𝑜 = suc (2𝑜 +𝑜 1𝑜)
134, 6, 123eqtr4i 2683 . 2 (2𝑜 +𝑜 2𝑜) = suc 3𝑜
14 df-4o 7608 . 2 4𝑜 = suc 3𝑜
1513, 14eqtr4i 2676 1 (2𝑜 +𝑜 2𝑜) = 4𝑜
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wcel 2030  Oncon0 5761  suc csuc 5763  (class class class)co 6690  1𝑜c1o 7598  2𝑜c2o 7599  3𝑜c3o 7600  4𝑜c4o 7601   +𝑜 coa 7602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-3o 7607  df-4o 7608  df-oadd 7609
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator