Theorem nnadd1com 42317

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 7412	. . 3 ⊢ (𝑥 = 1 → (𝑥 + 1) = (1 + 1))
2		oveq2 7413	. . 3 ⊢ (𝑥 = 1 → (1 + 𝑥) = (1 + 1))
3	1, 2	eqeq12d 2751	. 2 ⊢ (𝑥 = 1 → ((𝑥 + 1) = (1 + 𝑥) ↔ (1 + 1) = (1 + 1)))
4		oveq1 7412	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1))
5		oveq2 7413	. . 3 ⊢ (𝑥 = 𝑦 → (1 + 𝑥) = (1 + 𝑦))
6	4, 5	eqeq12d 2751	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 + 1) = (1 + 𝑥) ↔ (𝑦 + 1) = (1 + 𝑦)))
7		oveq1 7412	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1))
8		oveq2 7413	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (1 + 𝑥) = (1 + (𝑦 + 1)))
9	7, 8	eqeq12d 2751	. 2 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 + 1) = (1 + 𝑥) ↔ ((𝑦 + 1) + 1) = (1 + (𝑦 + 1))))
10		oveq1 7412	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 + 1) = (𝐴 + 1))
11		oveq2 7413	. . 3 ⊢ (𝑥 = 𝐴 → (1 + 𝑥) = (1 + 𝐴))
12	10, 11	eqeq12d 2751	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 + 1) = (1 + 𝑥) ↔ (𝐴 + 1) = (1 + 𝐴)))
13		eqid 2735	. 2 ⊢ (1 + 1) = (1 + 1)
14		oveq1 7412	. . . 4 ⊢ ((𝑦 + 1) = (1 + 𝑦) → ((𝑦 + 1) + 1) = ((1 + 𝑦) + 1))
15		1cnd 11230	. . . . 5 ⊢ (𝑦 ∈ ℕ → 1 ∈ ℂ)
16		nncn 12248	. . . . 5 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
17	15, 16, 15	addassd 11257	. . . 4 ⊢ (𝑦 ∈ ℕ → ((1 + 𝑦) + 1) = (1 + (𝑦 + 1)))
18	14, 17	sylan9eqr 2792	. . 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 12258	1 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) = (1 + 𝐴))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  (class class class)co 7405  1c1 11130   + caddc 11132  ℕcn 12240

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729  ax-1cn 11187  ax-addcl 11189  ax-addass 11194

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-om 7862  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-nn 12241

This theorem is referenced by:  nnaddcom  42318  renegmulnnass  42496