Theorem naddcom 8700

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 7431	. . 3 ⊢ (𝑎 = 𝑐 → (𝑎 +no 𝑏) = (𝑐 +no 𝑏))
2		oveq2 7432	. . 3 ⊢ (𝑎 = 𝑐 → (𝑏 +no 𝑎) = (𝑏 +no 𝑐))
3	1, 2	eqeq12d 2776	. 2 ⊢ (𝑎 = 𝑐 → ((𝑎 +no 𝑏) = (𝑏 +no 𝑎) ↔ (𝑐 +no 𝑏) = (𝑏 +no 𝑐)))
4		oveq2 7432	. . 3 ⊢ (𝑏 = 𝑑 → (𝑐 +no 𝑏) = (𝑐 +no 𝑑))
5		oveq1 7431	. . 3 ⊢ (𝑏 = 𝑑 → (𝑏 +no 𝑐) = (𝑑 +no 𝑐))
6	4, 5	eqeq12d 2776	. 2 ⊢ (𝑏 = 𝑑 → ((𝑐 +no 𝑏) = (𝑏 +no 𝑐) ↔ (𝑐 +no 𝑑) = (𝑑 +no 𝑐)))
7		oveq1 7431	. . 3 ⊢ (𝑎 = 𝑐 → (𝑎 +no 𝑑) = (𝑐 +no 𝑑))
8		oveq2 7432	. . 3 ⊢ (𝑎 = 𝑐 → (𝑑 +no 𝑎) = (𝑑 +no 𝑐))
9	7, 8	eqeq12d 2776	. 2 ⊢ (𝑎 = 𝑐 → ((𝑎 +no 𝑑) = (𝑑 +no 𝑎) ↔ (𝑐 +no 𝑑) = (𝑑 +no 𝑐)))
10		oveq1 7431	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 +no 𝑏) = (𝐴 +no 𝑏))
11		oveq2 7432	. . 3 ⊢ (𝑎 = 𝐴 → (𝑏 +no 𝑎) = (𝑏 +no 𝐴))
12	10, 11	eqeq12d 2776	. 2 ⊢ (𝑎 = 𝐴 → ((𝑎 +no 𝑏) = (𝑏 +no 𝑎) ↔ (𝐴 +no 𝑏) = (𝑏 +no 𝐴)))
13		oveq2 7432	. . 3 ⊢ (𝑏 = 𝐵 → (𝐴 +no 𝑏) = (𝐴 +no 𝐵))
14		oveq1 7431	. . 3 ⊢ (𝑏 = 𝐵 → (𝑏 +no 𝐴) = (𝐵 +no 𝐴))
15	13, 14	eqeq12d 2776	. 2 ⊢ (𝑏 = 𝐵 → ((𝐴 +no 𝑏) = (𝑏 +no 𝐴) ↔ (𝐴 +no 𝐵) = (𝐵 +no 𝐴)))
16		eleq1 2848	. . . . . . . . . . . 12 ⊢ ((𝑎 +no 𝑑) = (𝑑 +no 𝑎) → ((𝑎 +no 𝑑) ∈ 𝑥 ↔ (𝑑 +no 𝑎) ∈ 𝑥))
17	16	ralimi 3099	. . . . . . . . . . 11 ⊢ (∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎) → ∀𝑑 ∈ 𝑏 ((𝑎 +no 𝑑) ∈ 𝑥 ↔ (𝑑 +no 𝑎) ∈ 𝑥))
18		ralbi 3117	. . . . . . . . . . 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 2848	. . . . . . . . . . . 12 ⊢ ((𝑐 +no 𝑏) = (𝑏 +no 𝑐) → ((𝑐 +no 𝑏) ∈ 𝑥 ↔ (𝑏 +no 𝑐) ∈ 𝑥))
23	22	ralimi 3099	. . . . . . . . . . 11 ⊢ (∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) → ∀𝑐 ∈ 𝑎 ((𝑐 +no 𝑏) ∈ 𝑥 ↔ (𝑏 +no 𝑐) ∈ 𝑥))
24		ralbi 3117	. . . . . . . . . . 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 3419	. . . . 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 4911	. . . 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 8696	. . . . 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 8696	. . . . . 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 2805	. . 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 8686	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 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412  ∩ cint 4906  Oncon0 6365  (class class class)co 7424   +no cnadd 8682

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006  df-frecs 8307  df-nadd 8683

This theorem is used by:  naddlid  8702  naddel2  8706  naddss2  8708  naddword2  8710  nadd32  8715  nadd42  8717  addsproplem2  28365  addbday  28413  nmulcom  37067  naddcomd  37082  nmuladdss  37086  nadd2rabex  44525  nadd1rabtr  44527  nadd1rabex  44529  naddwordnexlem4  44540