Theorem naddcom 8612

Description: Natural addition commutes. (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 7368	. . 3 ⊢ (𝑎 = 𝑐 → (𝑎 +no 𝑏) = (𝑐 +no 𝑏))
2		oveq2 7369	. . 3 ⊢ (𝑎 = 𝑐 → (𝑏 +no 𝑎) = (𝑏 +no 𝑐))
3	1, 2	eqeq12d 2753	. 2 ⊢ (𝑎 = 𝑐 → ((𝑎 +no 𝑏) = (𝑏 +no 𝑎) ↔ (𝑐 +no 𝑏) = (𝑏 +no 𝑐)))
4		oveq2 7369	. . 3 ⊢ (𝑏 = 𝑑 → (𝑐 +no 𝑏) = (𝑐 +no 𝑑))
5		oveq1 7368	. . 3 ⊢ (𝑏 = 𝑑 → (𝑏 +no 𝑐) = (𝑑 +no 𝑐))
6	4, 5	eqeq12d 2753	. 2 ⊢ (𝑏 = 𝑑 → ((𝑐 +no 𝑏) = (𝑏 +no 𝑐) ↔ (𝑐 +no 𝑑) = (𝑑 +no 𝑐)))
7		oveq1 7368	. . 3 ⊢ (𝑎 = 𝑐 → (𝑎 +no 𝑑) = (𝑐 +no 𝑑))
8		oveq2 7369	. . 3 ⊢ (𝑎 = 𝑐 → (𝑑 +no 𝑎) = (𝑑 +no 𝑐))
9	7, 8	eqeq12d 2753	. 2 ⊢ (𝑎 = 𝑐 → ((𝑎 +no 𝑑) = (𝑑 +no 𝑎) ↔ (𝑐 +no 𝑑) = (𝑑 +no 𝑐)))
10		oveq1 7368	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 +no 𝑏) = (𝐴 +no 𝑏))
11		oveq2 7369	. . 3 ⊢ (𝑎 = 𝐴 → (𝑏 +no 𝑎) = (𝑏 +no 𝐴))
12	10, 11	eqeq12d 2753	. 2 ⊢ (𝑎 = 𝐴 → ((𝑎 +no 𝑏) = (𝑏 +no 𝑎) ↔ (𝐴 +no 𝑏) = (𝑏 +no 𝐴)))
13		oveq2 7369	. . 3 ⊢ (𝑏 = 𝐵 → (𝐴 +no 𝑏) = (𝐴 +no 𝐵))
14		oveq1 7368	. . 3 ⊢ (𝑏 = 𝐵 → (𝑏 +no 𝐴) = (𝐵 +no 𝐴))
15	13, 14	eqeq12d 2753	. 2 ⊢ (𝑏 = 𝐵 → ((𝐴 +no 𝑏) = (𝑏 +no 𝐴) ↔ (𝐴 +no 𝐵) = (𝐵 +no 𝐴)))
16		eleq1 2825	. . . . . . . . . . . 12 ⊢ ((𝑎 +no 𝑑) = (𝑑 +no 𝑎) → ((𝑎 +no 𝑑) ∈ 𝑥 ↔ (𝑑 +no 𝑎) ∈ 𝑥))
17	16	ralimi 3075	. . . . . . . . . . 11 ⊢ (∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎) → ∀𝑑 ∈ 𝑏 ((𝑎 +no 𝑑) ∈ 𝑥 ↔ (𝑑 +no 𝑎) ∈ 𝑥))
18		ralbi 3093	. . . . . . . . . . 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 17	. . . . . . . . . 10 ⊢ (∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎) → (∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ↔ ∀𝑑 ∈ 𝑏 (𝑑 +no 𝑎) ∈ 𝑥))
20	19	3ad2ant3 1136	. . . . . . . . 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 481	. . . . . . . 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 2825	. . . . . . . . . . . 12 ⊢ ((𝑐 +no 𝑏) = (𝑏 +no 𝑐) → ((𝑐 +no 𝑏) ∈ 𝑥 ↔ (𝑏 +no 𝑐) ∈ 𝑥))
23	22	ralimi 3075	. . . . . . . . . . 11 ⊢ (∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) → ∀𝑐 ∈ 𝑎 ((𝑐 +no 𝑏) ∈ 𝑥 ↔ (𝑏 +no 𝑐) ∈ 𝑥))
24		ralbi 3093	. . . . . . . . . . 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 17	. . . . . . . . . 10 ⊢ (∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) → (∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) ∈ 𝑥 ↔ ∀𝑐 ∈ 𝑎 (𝑏 +no 𝑐) ∈ 𝑥))
26	25	3ad2ant2 1135	. . . . . . . . 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 481	. . . . . . . 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 633	. . . . . . 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 463	. . . . . 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 3397	. . . . 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 4895	. . . 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 8609	. . . . 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 480	. . . 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 8609	. . . . . 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 458	. . . . 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 480	. . . 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 2782	. . 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 412	. 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 8599	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = (𝐵 +no 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3390  ∩ cint 4890  Oncon0 6318  (class class class)co 7361   +no cnadd 8595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7936  df-2nd 7937  df-frecs 8225  df-nadd 8596

This theorem is referenced by:  naddlid  8614  naddel2  8618  naddss2  8620  naddword2  8622  nadd32  8627  nadd42  8629  addsproplem2  27979  addbday  28027  nadd2rabex  43835  nadd1rabtr  43837  nadd1rabex  43839  naddwordnexlem4  43850