Theorem o2p2e4 7508

Description: 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 5646. For the usual proof using complex numbers, see 2p2e4 11021. (Contributed by NM, 18-Aug-2021.)

Assertion

Ref	Expression
o2p2e4	⊢ (2𝑜 +𝑜 2𝑜) = 4𝑜

Proof of Theorem o2p2e4

Step	Hyp	Ref	Expression
1		2on 7455	. . . 4 ⊢ 2𝑜 ∈ On
2		1on 7454	. . . 4 ⊢ 1𝑜 ∈ On
3		oasuc 7491	. . . 4 ⊢ ((2𝑜 ∈ On ∧ 1𝑜 ∈ On) → (2𝑜 +𝑜 suc 1𝑜) = suc (2𝑜 +𝑜 1𝑜))
4	1, 2, 3	mp2an 704	. . 3 ⊢ (2𝑜 +𝑜 suc 1𝑜) = suc (2𝑜 +𝑜 1𝑜)
5		df-2o 7448	. . . 4 ⊢ 2𝑜 = suc 1𝑜
6	5	oveq2i 6560	. . 3 ⊢ (2𝑜 +𝑜 2𝑜) = (2𝑜 +𝑜 suc 1𝑜)
7		df-3o 7449	. . . . 5 ⊢ 3𝑜 = suc 2𝑜
8		oa1suc 7498	. . . . . 6 ⊢ (2𝑜 ∈ On → (2𝑜 +𝑜 1𝑜) = suc 2𝑜)
9	1, 8	ax-mp 5	. . . . 5 ⊢ (2𝑜 +𝑜 1𝑜) = suc 2𝑜
10	7, 9	eqtr4i 2635	. . . 4 ⊢ 3𝑜 = (2𝑜 +𝑜 1𝑜)
11		suceq 5707	. . . 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 2642	. 2 ⊢ (2𝑜 +𝑜 2𝑜) = suc 3𝑜
14		df-4o 7450	. 2 ⊢ 4𝑜 = suc 3𝑜
15	13, 14	eqtr4i 2635	1 ⊢ (2𝑜 +𝑜 2𝑜) = 4𝑜

Syntax hints:   = wceq 1475   ∈ wcel 1977  Oncon0 5640  suc csuc 5642  (class class class)co 6549  1_𝑜c1o 7440  2_𝑜c2o 7441  3_𝑜c3o 7442  4_𝑜c4o 7443   +_𝑜 coa 7444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-3o 7449  df-4o 7450  df-oadd 7451

This theorem is referenced by: (None)