Theorem nnaddcom 41736

Description: Addition is commutative for natural numbers. Uses fewer axioms than addcom 11404. (Contributed by Steven Nguyen, 9-Dec-2022.)

Assertion

Ref	Expression
nnaddcom	⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Proof of Theorem nnaddcom

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7412	. . . . 5 ⊢ (𝑥 = 1 → (𝑥 + 𝐵) = (1 + 𝐵))
2		oveq2 7413	. . . . 5 ⊢ (𝑥 = 1 → (𝐵 + 𝑥) = (𝐵 + 1))
3	1, 2	eqeq12d 2742	. . . 4 ⊢ (𝑥 = 1 → ((𝑥 + 𝐵) = (𝐵 + 𝑥) ↔ (1 + 𝐵) = (𝐵 + 1)))
4	3	imbi2d 340	. . 3 ⊢ (𝑥 = 1 → ((𝐵 ∈ ℕ → (𝑥 + 𝐵) = (𝐵 + 𝑥)) ↔ (𝐵 ∈ ℕ → (1 + 𝐵) = (𝐵 + 1))))
5		oveq1 7412	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 + 𝐵) = (𝑦 + 𝐵))
6		oveq2 7413	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 + 𝑥) = (𝐵 + 𝑦))
7	5, 6	eqeq12d 2742	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 + 𝐵) = (𝐵 + 𝑥) ↔ (𝑦 + 𝐵) = (𝐵 + 𝑦)))
8	7	imbi2d 340	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐵 ∈ ℕ → (𝑥 + 𝐵) = (𝐵 + 𝑥)) ↔ (𝐵 ∈ ℕ → (𝑦 + 𝐵) = (𝐵 + 𝑦))))
9		oveq1 7412	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 + 𝐵) = ((𝑦 + 1) + 𝐵))
10		oveq2 7413	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝐵 + 𝑥) = (𝐵 + (𝑦 + 1)))
11	9, 10	eqeq12d 2742	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 + 𝐵) = (𝐵 + 𝑥) ↔ ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1))))
12	11	imbi2d 340	. . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝐵 ∈ ℕ → (𝑥 + 𝐵) = (𝐵 + 𝑥)) ↔ (𝐵 ∈ ℕ → ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1)))))
13		oveq1 7412	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 + 𝐵) = (𝐴 + 𝐵))
14		oveq2 7413	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 + 𝑥) = (𝐵 + 𝐴))
15	13, 14	eqeq12d 2742	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 + 𝐵) = (𝐵 + 𝑥) ↔ (𝐴 + 𝐵) = (𝐵 + 𝐴)))
16	15	imbi2d 340	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ ℕ → (𝑥 + 𝐵) = (𝐵 + 𝑥)) ↔ (𝐵 ∈ ℕ → (𝐴 + 𝐵) = (𝐵 + 𝐴))))
17		nnadd1com 41735	. . . 4 ⊢ (𝐵 ∈ ℕ → (𝐵 + 1) = (1 + 𝐵))
18	17	eqcomd 2732	. . 3 ⊢ (𝐵 ∈ ℕ → (1 + 𝐵) = (𝐵 + 1))
19		oveq1 7412	. . . . . 6 ⊢ ((𝑦 + 𝐵) = (𝐵 + 𝑦) → ((𝑦 + 𝐵) + 1) = ((𝐵 + 𝑦) + 1))
20	17	oveq2d 7421	. . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → (𝑦 + (𝐵 + 1)) = (𝑦 + (1 + 𝐵)))
21	20	adantl 481	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑦 + (𝐵 + 1)) = (𝑦 + (1 + 𝐵)))
22		nncn 12224	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
23	22	adantr 480	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝑦 ∈ ℂ)
24		nncn 12224	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
25	24	adantl 481	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℂ)
26		1cnd 11213	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 1 ∈ ℂ)
27	23, 25, 26	addassd 11240	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝑦 + 𝐵) + 1) = (𝑦 + (𝐵 + 1)))
28	23, 26, 25	addassd 11240	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝑦 + 1) + 𝐵) = (𝑦 + (1 + 𝐵)))
29	21, 27, 28	3eqtr4d 2776	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝑦 + 𝐵) + 1) = ((𝑦 + 1) + 𝐵))
30	25, 23, 26	addassd 11240	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐵 + 𝑦) + 1) = (𝐵 + (𝑦 + 1)))
31	29, 30	eqeq12d 2742	. . . . . 6 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (((𝑦 + 𝐵) + 1) = ((𝐵 + 𝑦) + 1) ↔ ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1))))
32	19, 31	imbitrid 243	. . . . 5 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝑦 + 𝐵) = (𝐵 + 𝑦) → ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1))))
33	32	ex 412	. . . 4 ⊢ (𝑦 ∈ ℕ → (𝐵 ∈ ℕ → ((𝑦 + 𝐵) = (𝐵 + 𝑦) → ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1)))))
34	33	a2d 29	. . 3 ⊢ (𝑦 ∈ ℕ → ((𝐵 ∈ ℕ → (𝑦 + 𝐵) = (𝐵 + 𝑦)) → (𝐵 ∈ ℕ → ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1)))))
35	4, 8, 12, 16, 18, 34	nnind 12234	. 2 ⊢ (𝐴 ∈ ℕ → (𝐵 ∈ ℕ → (𝐴 + 𝐵) = (𝐵 + 𝐴)))
36	35	imp 406	1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  (class class class)co 7405  ℂcc 11110  1c1 11113   + caddc 11115  ℕcn 12216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722  ax-1cn 11170  ax-addcl 11172  ax-addass 11177

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-om 7853  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-nn 12217

This theorem is referenced by:  nnaddcomli  41737  nnadddir  41738  nn0addcom  41898  zaddcom  41900