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

Step	Hyp	Ref	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))
4	1, 2, 3	mp2an 708	. . 3 ⊢ (2𝑜 +𝑜 suc 1𝑜) = suc (2𝑜 +𝑜 1𝑜)
5		df-2o 7606	. . . 4 ⊢ 2𝑜 = suc 1𝑜
6	5	oveq2i 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)
9	1, 8	ax-mp 5	. . . . 5 ⊢ (2𝑜 +𝑜 1𝑜) = suc 2𝑜
10	7, 9	eqtr4i 2676	. . . 4 ⊢ 3𝑜 = (2𝑜 +𝑜 1𝑜)
11		suceq 5828	. . . 4 ⊢ (3𝑜 = (2𝑜 +𝑜 1𝑜) → suc 3𝑜 = suc (2𝑜 +𝑜 1𝑜))
12	10, 11	ax-mp 5	. . 3 ⊢ suc 3𝑜 = suc (2𝑜 +𝑜 1𝑜)
13	4, 6, 12	3eqtr4i 2683	. 2 ⊢ (2𝑜 +𝑜 2𝑜) = suc 3𝑜
14		df-4o 7608	. 2 ⊢ 4𝑜 = suc 3𝑜
15	13, 14	eqtr4i 2676	1 ⊢ (2𝑜 +𝑜 2𝑜) = 4𝑜

Syntax hints:   = wceq 1523   ∈ wcel 2030  Oncon0 5761  suc csuc 5763  (class class class)co 6690  1_𝑜c1o 7598  2_𝑜c2o 7599  3_𝑜c3o 7600  4_𝑜c4o 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)