Theorem nnaddcom 40770

Description: Addition is commutative for natural numbers. Uses fewer axioms than addcom 11341. (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 7364	. . . . 5 ⊢ (𝑥 = 1 → (𝑥 + 𝐵) = (1 + 𝐵))
2		oveq2 7365	. . . . 5 ⊢ (𝑥 = 1 → (𝐵 + 𝑥) = (𝐵 + 1))
3	1, 2	eqeq12d 2752	. . . 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 7364	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 + 𝐵) = (𝑦 + 𝐵))
6		oveq2 7365	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 + 𝑥) = (𝐵 + 𝑦))
7	5, 6	eqeq12d 2752	. . . 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 7364	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 + 𝐵) = ((𝑦 + 1) + 𝐵))
10		oveq2 7365	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝐵 + 𝑥) = (𝐵 + (𝑦 + 1)))
11	9, 10	eqeq12d 2752	. . . 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 7364	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 + 𝐵) = (𝐴 + 𝐵))
14		oveq2 7365	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 + 𝑥) = (𝐵 + 𝐴))
15	13, 14	eqeq12d 2752	. . . 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 40769	. . . 4 ⊢ (𝐵 ∈ ℕ → (𝐵 + 1) = (1 + 𝐵))
18	17	eqcomd 2742	. . 3 ⊢ (𝐵 ∈ ℕ → (1 + 𝐵) = (𝐵 + 1))
19		oveq1 7364	. . . . . 6 ⊢ ((𝑦 + 𝐵) = (𝐵 + 𝑦) → ((𝑦 + 𝐵) + 1) = ((𝐵 + 𝑦) + 1))
20	17	oveq2d 7373	. . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → (𝑦 + (𝐵 + 1)) = (𝑦 + (1 + 𝐵)))
21	20	adantl 482	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑦 + (𝐵 + 1)) = (𝑦 + (1 + 𝐵)))
22		nncn 12161	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
23	22	adantr 481	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝑦 ∈ ℂ)
24		nncn 12161	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
25	24	adantl 482	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℂ)
26		1cnd 11150	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 1 ∈ ℂ)
27	23, 25, 26	addassd 11177	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝑦 + 𝐵) + 1) = (𝑦 + (𝐵 + 1)))
28	23, 26, 25	addassd 11177	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝑦 + 1) + 𝐵) = (𝑦 + (1 + 𝐵)))
29	21, 27, 28	3eqtr4d 2786	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝑦 + 𝐵) + 1) = ((𝑦 + 1) + 𝐵))
30	25, 23, 26	addassd 11177	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐵 + 𝑦) + 1) = (𝐵 + (𝑦 + 1)))
31	29, 30	eqeq12d 2752	. . . . . 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 413	. . . 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 12171	. 2 ⊢ (𝐴 ∈ ℕ → (𝐵 ∈ ℕ → (𝐴 + 𝐵) = (𝐵 + 𝐴)))
36	35	imp 407	1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  (class class class)co 7357  ℂcc 11049  1c1 11052   + caddc 11054  ℕcn 12153

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672  ax-1cn 11109  ax-addcl 11111  ax-addass 11116

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-nn 12154

This theorem is referenced by:  nnaddcomli  40771  nnadddir  40772  nn0addcom  40905  zaddcom  40907