Theorem nnaddcom 12259

Description: Addition is commutative for natural numbers. Uses fewer axioms than addcom 11395. (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 7418	. . . . 5 ⊢ (𝑥 = 1 → (𝑥 + 𝐵) = (1 + 𝐵))
2		oveq2 7419	. . . . 5 ⊢ (𝑥 = 1 → (𝐵 + 𝑥) = (𝐵 + 1))
3	1, 2	eqeq12d 2785	. . . 4 ⊢ (𝑥 = 1 → ((𝑥 + 𝐵) = (𝐵 + 𝑥) ↔ (1 + 𝐵) = (𝐵 + 1)))
4	3	imbi2d 343	. . 3 ⊢ (𝑥 = 1 → ((𝐵 ∈ ℕ → (𝑥 + 𝐵) = (𝐵 + 𝑥)) ↔ (𝐵 ∈ ℕ → (1 + 𝐵) = (𝐵 + 1))))
5		oveq1 7418	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 + 𝐵) = (𝑦 + 𝐵))
6		oveq2 7419	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 + 𝑥) = (𝐵 + 𝑦))
7	5, 6	eqeq12d 2785	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 + 𝐵) = (𝐵 + 𝑥) ↔ (𝑦 + 𝐵) = (𝐵 + 𝑦)))
8	7	imbi2d 343	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐵 ∈ ℕ → (𝑥 + 𝐵) = (𝐵 + 𝑥)) ↔ (𝐵 ∈ ℕ → (𝑦 + 𝐵) = (𝐵 + 𝑦))))
9		oveq1 7418	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 + 𝐵) = ((𝑦 + 1) + 𝐵))
10		oveq2 7419	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝐵 + 𝑥) = (𝐵 + (𝑦 + 1)))
11	9, 10	eqeq12d 2785	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 + 𝐵) = (𝐵 + 𝑥) ↔ ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1))))
12	11	imbi2d 343	. . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝐵 ∈ ℕ → (𝑥 + 𝐵) = (𝐵 + 𝑥)) ↔ (𝐵 ∈ ℕ → ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1)))))
13		oveq1 7418	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 + 𝐵) = (𝐴 + 𝐵))
14		oveq2 7419	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 + 𝑥) = (𝐵 + 𝐴))
15	13, 14	eqeq12d 2785	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 + 𝐵) = (𝐵 + 𝑥) ↔ (𝐴 + 𝐵) = (𝐵 + 𝐴)))
16	15	imbi2d 343	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ ℕ → (𝑥 + 𝐵) = (𝐵 + 𝑥)) ↔ (𝐵 ∈ ℕ → (𝐴 + 𝐵) = (𝐵 + 𝐴))))
17		nnadd1com 12258	. . . 4 ⊢ (𝐵 ∈ ℕ → (𝐵 + 1) = (1 + 𝐵))
18	17	eqcomd 2775	. . 3 ⊢ (𝐵 ∈ ℕ → (1 + 𝐵) = (𝐵 + 1))
19		oveq1 7418	. . . . . 6 ⊢ ((𝑦 + 𝐵) = (𝐵 + 𝑦) → ((𝑦 + 𝐵) + 1) = ((𝐵 + 𝑦) + 1))
20	17	oveq2d 7427	. . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → (𝑦 + (𝐵 + 1)) = (𝑦 + (1 + 𝐵)))
21	20	adantl 486	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑦 + (𝐵 + 1)) = (𝑦 + (1 + 𝐵)))
22		nncn 12240	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
23	22	adantr 485	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝑦 ∈ ℂ)
24		nncn 12240	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
25	24	adantl 486	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℂ)
26		1cnd 11201	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 1 ∈ ℂ)
27	23, 25, 26	addassd 11230	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝑦 + 𝐵) + 1) = (𝑦 + (𝐵 + 1)))
28	23, 26, 25	addassd 11230	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝑦 + 1) + 𝐵) = (𝑦 + (1 + 𝐵)))
29	21, 27, 28	3eqtr4d 2814	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝑦 + 𝐵) + 1) = ((𝑦 + 1) + 𝐵))
30	25, 23, 26	addassd 11230	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐵 + 𝑦) + 1) = (𝐵 + (𝑦 + 1)))
31	29, 30	eqeq12d 2785	. . . . . 6 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (((𝑦 + 𝐵) + 1) = ((𝐵 + 𝑦) + 1) ↔ ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1))))
32	19, 31	imbitrid 247	. . . . 5 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝑦 + 𝐵) = (𝐵 + 𝑦) → ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1))))
33	32	ex 417	. . . 4 ⊢ (𝑦 ∈ ℕ → (𝐵 ∈ ℕ → ((𝑦 + 𝐵) = (𝐵 + 𝑦) → ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1)))))
34	33	a2d 30	. . 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 12250	. 2 ⊢ (𝐴 ∈ ℕ → (𝐵 ∈ ℕ → (𝐴 + 𝐵) = (𝐵 + 𝐴)))
36	35	imp 411	1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  (class class class)co 7411  ℂcc 11097  1c1 11100   + caddc 11102  ℕcn 12232

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733  ax-1cn 11157  ax-addcl 11159  ax-addass 11164

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-nn 12233

This theorem is referenced by:  nnaddcomli  12260  nnadddir  12291  nn0addcom  43125  zaddcom  43127