Theorem naddcom 8675

Description: Natural addition is commutative. (Contributed by Scott Fenton, 26-Aug-2024.)

Assertion

Ref	Expression
naddcom	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = (𝐵 +no 𝐴))

Proof of Theorem naddcom

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7426	. . 3 ⊢ (𝑎 = 𝑐 → (𝑎 +no 𝑏) = (𝑐 +no 𝑏))
2		oveq2 7427	. . 3 ⊢ (𝑎 = 𝑐 → (𝑏 +no 𝑎) = (𝑏 +no 𝑐))
3	1, 2	eqeq12d 2781	. 2 ⊢ (𝑎 = 𝑐 → ((𝑎 +no 𝑏) = (𝑏 +no 𝑎) ↔ (𝑐 +no 𝑏) = (𝑏 +no 𝑐)))
4		oveq2 7427	. . 3 ⊢ (𝑏 = 𝑑 → (𝑐 +no 𝑏) = (𝑐 +no 𝑑))
5		oveq1 7426	. . 3 ⊢ (𝑏 = 𝑑 → (𝑏 +no 𝑐) = (𝑑 +no 𝑐))
6	4, 5	eqeq12d 2781	. 2 ⊢ (𝑏 = 𝑑 → ((𝑐 +no 𝑏) = (𝑏 +no 𝑐) ↔ (𝑐 +no 𝑑) = (𝑑 +no 𝑐)))
7		oveq1 7426	. . 3 ⊢ (𝑎 = 𝑐 → (𝑎 +no 𝑑) = (𝑐 +no 𝑑))
8		oveq2 7427	. . 3 ⊢ (𝑎 = 𝑐 → (𝑑 +no 𝑎) = (𝑑 +no 𝑐))
9	7, 8	eqeq12d 2781	. 2 ⊢ (𝑎 = 𝑐 → ((𝑎 +no 𝑑) = (𝑑 +no 𝑎) ↔ (𝑐 +no 𝑑) = (𝑑 +no 𝑐)))
10		oveq1 7426	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 +no 𝑏) = (𝐴 +no 𝑏))
11		oveq2 7427	. . 3 ⊢ (𝑎 = 𝐴 → (𝑏 +no 𝑎) = (𝑏 +no 𝐴))
12	10, 11	eqeq12d 2781	. 2 ⊢ (𝑎 = 𝐴 → ((𝑎 +no 𝑏) = (𝑏 +no 𝑎) ↔ (𝐴 +no 𝑏) = (𝑏 +no 𝐴)))
13		oveq2 7427	. . 3 ⊢ (𝑏 = 𝐵 → (𝐴 +no 𝑏) = (𝐴 +no 𝐵))
14		oveq1 7426	. . 3 ⊢ (𝑏 = 𝐵 → (𝑏 +no 𝐴) = (𝐵 +no 𝐴))
15	13, 14	eqeq12d 2781	. 2 ⊢ (𝑏 = 𝐵 → ((𝐴 +no 𝑏) = (𝑏 +no 𝐴) ↔ (𝐴 +no 𝐵) = (𝐵 +no 𝐴)))
16		eleq1 2853	. . . . . . . . . . . 12 ⊢ ((𝑎 +no 𝑑) = (𝑑 +no 𝑎) → ((𝑎 +no 𝑑) ∈ 𝑥 ↔ (𝑑 +no 𝑎) ∈ 𝑥))
17	16	ralimi 3104	. . . . . . . . . . 11 ⊢ (∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎) → ∀𝑑 ∈ 𝑏 ((𝑎 +no 𝑑) ∈ 𝑥 ↔ (𝑑 +no 𝑎) ∈ 𝑥))
18		ralbi 3122	. . . . . . . . . . 11 ⊢ (∀𝑑 ∈ 𝑏 ((𝑎 +no 𝑑) ∈ 𝑥 ↔ (𝑑 +no 𝑎) ∈ 𝑥) → (∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ↔ ∀𝑑 ∈ 𝑏 (𝑑 +no 𝑎) ∈ 𝑥))
19	17, 18	syl 18	. . . . . . . . . 10 ⊢ (∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎) → (∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ↔ ∀𝑑 ∈ 𝑏 (𝑑 +no 𝑎) ∈ 𝑥))
20	19	3ad2ant3 1153	. . . . . . . . 9 ⊢ ((∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎)) → (∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ↔ ∀𝑑 ∈ 𝑏 (𝑑 +no 𝑎) ∈ 𝑥))
21	20	adantl 487	. . . . . . . 8 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → (∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ↔ ∀𝑑 ∈ 𝑏 (𝑑 +no 𝑎) ∈ 𝑥))
22		eleq1 2853	. . . . . . . . . . . 12 ⊢ ((𝑐 +no 𝑏) = (𝑏 +no 𝑐) → ((𝑐 +no 𝑏) ∈ 𝑥 ↔ (𝑏 +no 𝑐) ∈ 𝑥))
23	22	ralimi 3104	. . . . . . . . . . 11 ⊢ (∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) → ∀𝑐 ∈ 𝑎 ((𝑐 +no 𝑏) ∈ 𝑥 ↔ (𝑏 +no 𝑐) ∈ 𝑥))
24		ralbi 3122	. . . . . . . . . . 11 ⊢ (∀𝑐 ∈ 𝑎 ((𝑐 +no 𝑏) ∈ 𝑥 ↔ (𝑏 +no 𝑐) ∈ 𝑥) → (∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) ∈ 𝑥 ↔ ∀𝑐 ∈ 𝑎 (𝑏 +no 𝑐) ∈ 𝑥))
25	23, 24	syl 18	. . . . . . . . . 10 ⊢ (∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) → (∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) ∈ 𝑥 ↔ ∀𝑐 ∈ 𝑎 (𝑏 +no 𝑐) ∈ 𝑥))
26	25	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎)) → (∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) ∈ 𝑥 ↔ ∀𝑐 ∈ 𝑎 (𝑏 +no 𝑐) ∈ 𝑥))
27	26	adantl 487	. . . . . . . 8 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → (∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) ∈ 𝑥 ↔ ∀𝑐 ∈ 𝑎 (𝑏 +no 𝑐) ∈ 𝑥))
28	21, 27	anbi12d 644	. . . . . . 7 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → ((∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) ∈ 𝑥) ↔ (∀𝑑 ∈ 𝑏 (𝑑 +no 𝑎) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝑎 (𝑏 +no 𝑐) ∈ 𝑥)))
29	28	biancomd 469	. . . . . 6 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → ((∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) ∈ 𝑥) ↔ (∀𝑐 ∈ 𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑 ∈ 𝑏 (𝑑 +no 𝑎) ∈ 𝑥)))
30	29	rabbidv 3425	. . . . 5 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → {𝑥 ∈ On ∣ (∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) ∈ 𝑥)} = {𝑥 ∈ On ∣ (∀𝑐 ∈ 𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑 ∈ 𝑏 (𝑑 +no 𝑎) ∈ 𝑥)})
31	30	inteqd 4919	. . . 4 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → ∩ {𝑥 ∈ On ∣ (∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) ∈ 𝑥)} = ∩ {𝑥 ∈ On ∣ (∀𝑐 ∈ 𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑 ∈ 𝑏 (𝑑 +no 𝑎) ∈ 𝑥)})
32		naddov2 8671	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 +no 𝑏) = ∩ {𝑥 ∈ On ∣ (∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) ∈ 𝑥)})
33	32	adantr 486	. . . 4 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → (𝑎 +no 𝑏) = ∩ {𝑥 ∈ On ∣ (∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) ∈ 𝑥)})
34		naddov2 8671	. . . . . 6 ⊢ ((𝑏 ∈ On ∧ 𝑎 ∈ On) → (𝑏 +no 𝑎) = ∩ {𝑥 ∈ On ∣ (∀𝑐 ∈ 𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑 ∈ 𝑏 (𝑑 +no 𝑎) ∈ 𝑥)})
35	34	ancoms 464	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑏 +no 𝑎) = ∩ {𝑥 ∈ On ∣ (∀𝑐 ∈ 𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑 ∈ 𝑏 (𝑑 +no 𝑎) ∈ 𝑥)})
36	35	adantr 486	. . . 4 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → (𝑏 +no 𝑎) = ∩ {𝑥 ∈ On ∣ (∀𝑐 ∈ 𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑 ∈ 𝑏 (𝑑 +no 𝑎) ∈ 𝑥)})
37	31, 33, 36	3eqtr4d 2810	. . 3 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → (𝑎 +no 𝑏) = (𝑏 +no 𝑎))
38	37	ex 418	. 2 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎)) → (𝑎 +no 𝑏) = (𝑏 +no 𝑎)))
39	3, 6, 9, 12, 15, 38	on2ind 8661	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = (𝐵 +no 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081  {crab 3418  ∩ cint 4914  Oncon0 6364  (class class class)co 7419   +no cnadd 8657

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-frecs 8284  df-nadd 8658

This theorem is used by:  naddlid  8677  naddel2  8681  naddss2  8683  naddword2  8685  nadd32  8690  nadd42  8692  addsproplem2  28216  addbday  28264  nmulcom  36725  naddcomd  36740  nmuladdss  36744  nadd2rabex  44173  nadd1rabtr  44175  nadd1rabex  44177  naddwordnexlem4  44188