Theorem nnadd1com 42244

Description: Addition with 1 is commutative for natural numbers. (Contributed by Steven Nguyen, 9-Dec-2022.)

Assertion

Ref	Expression
nnadd1com	⊢ (𝐴 ∈ ℕ → (𝐴 + 1) = (1 + 𝐴))

Proof of Theorem nnadd1com

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

Step	Hyp	Ref	Expression
1		oveq1 7356	. . 3 ⊢ (𝑥 = 1 → (𝑥 + 1) = (1 + 1))
2		oveq2 7357	. . 3 ⊢ (𝑥 = 1 → (1 + 𝑥) = (1 + 1))
3	1, 2	eqeq12d 2745	. 2 ⊢ (𝑥 = 1 → ((𝑥 + 1) = (1 + 𝑥) ↔ (1 + 1) = (1 + 1)))
4		oveq1 7356	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1))
5		oveq2 7357	. . 3 ⊢ (𝑥 = 𝑦 → (1 + 𝑥) = (1 + 𝑦))
6	4, 5	eqeq12d 2745	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 + 1) = (1 + 𝑥) ↔ (𝑦 + 1) = (1 + 𝑦)))
7		oveq1 7356	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1))
8		oveq2 7357	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (1 + 𝑥) = (1 + (𝑦 + 1)))
9	7, 8	eqeq12d 2745	. 2 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 + 1) = (1 + 𝑥) ↔ ((𝑦 + 1) + 1) = (1 + (𝑦 + 1))))
10		oveq1 7356	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 + 1) = (𝐴 + 1))
11		oveq2 7357	. . 3 ⊢ (𝑥 = 𝐴 → (1 + 𝑥) = (1 + 𝐴))
12	10, 11	eqeq12d 2745	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 + 1) = (1 + 𝑥) ↔ (𝐴 + 1) = (1 + 𝐴)))
13		eqid 2729	. 2 ⊢ (1 + 1) = (1 + 1)
14		oveq1 7356	. . . 4 ⊢ ((𝑦 + 1) = (1 + 𝑦) → ((𝑦 + 1) + 1) = ((1 + 𝑦) + 1))
15		1cnd 11110	. . . . 5 ⊢ (𝑦 ∈ ℕ → 1 ∈ ℂ)
16		nncn 12136	. . . . 5 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
17	15, 16, 15	addassd 11137	. . . 4 ⊢ (𝑦 ∈ ℕ → ((1 + 𝑦) + 1) = (1 + (𝑦 + 1)))
18	14, 17	sylan9eqr 2786	. . 3 ⊢ ((𝑦 ∈ ℕ ∧ (𝑦 + 1) = (1 + 𝑦)) → ((𝑦 + 1) + 1) = (1 + (𝑦 + 1)))
19	18	ex 412	. 2 ⊢ (𝑦 ∈ ℕ → ((𝑦 + 1) = (1 + 𝑦) → ((𝑦 + 1) + 1) = (1 + (𝑦 + 1))))
20	3, 6, 9, 12, 13, 19	nnind 12146	1 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) = (1 + 𝐴))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  (class class class)co 7349  1c1 11010   + caddc 11012  ℕcn 12128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671  ax-1cn 11067  ax-addcl 11069  ax-addass 11074

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-nn 12129

This theorem is referenced by:  nnaddcom  42245  renegmulnnass  42442