Theorem nnadd1com 39209

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 7163	. . 3 ⊢ (𝑥 = 1 → (𝑥 + 1) = (1 + 1))
2		oveq2 7164	. . 3 ⊢ (𝑥 = 1 → (1 + 𝑥) = (1 + 1))
3	1, 2	eqeq12d 2837	. 2 ⊢ (𝑥 = 1 → ((𝑥 + 1) = (1 + 𝑥) ↔ (1 + 1) = (1 + 1)))
4		oveq1 7163	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1))
5		oveq2 7164	. . 3 ⊢ (𝑥 = 𝑦 → (1 + 𝑥) = (1 + 𝑦))
6	4, 5	eqeq12d 2837	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 + 1) = (1 + 𝑥) ↔ (𝑦 + 1) = (1 + 𝑦)))
7		oveq1 7163	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1))
8		oveq2 7164	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (1 + 𝑥) = (1 + (𝑦 + 1)))
9	7, 8	eqeq12d 2837	. 2 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 + 1) = (1 + 𝑥) ↔ ((𝑦 + 1) + 1) = (1 + (𝑦 + 1))))
10		oveq1 7163	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 + 1) = (𝐴 + 1))
11		oveq2 7164	. . 3 ⊢ (𝑥 = 𝐴 → (1 + 𝑥) = (1 + 𝐴))
12	10, 11	eqeq12d 2837	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 + 1) = (1 + 𝑥) ↔ (𝐴 + 1) = (1 + 𝐴)))
13		eqid 2821	. 2 ⊢ (1 + 1) = (1 + 1)
14		oveq1 7163	. . . 4 ⊢ ((𝑦 + 1) = (1 + 𝑦) → ((𝑦 + 1) + 1) = ((1 + 𝑦) + 1))
15		1cnd 10636	. . . . 5 ⊢ (𝑦 ∈ ℕ → 1 ∈ ℂ)
16		nncn 11646	. . . . 5 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
17	15, 16, 15	addassd 10663	. . . 4 ⊢ (𝑦 ∈ ℕ → ((1 + 𝑦) + 1) = (1 + (𝑦 + 1)))
18	14, 17	sylan9eqr 2878	. . 3 ⊢ ((𝑦 ∈ ℕ ∧ (𝑦 + 1) = (1 + 𝑦)) → ((𝑦 + 1) + 1) = (1 + (𝑦 + 1)))
19	18	ex 415	. 2 ⊢ (𝑦 ∈ ℕ → ((𝑦 + 1) = (1 + 𝑦) → ((𝑦 + 1) + 1) = (1 + (𝑦 + 1))))
20	3, 6, 9, 12, 13, 19	nnind 11656	1 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) = (1 + 𝐴))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  (class class class)co 7156  1c1 10538   + caddc 10540  ℕcn 11638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-1cn 10595  ax-addcl 10597  ax-addass 10602

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-om 7581  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-nn 11639

This theorem is referenced by:  nnaddcom  39210