Theorem nnadd1com 42262

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 7397	. . 3 ⊢ (𝑥 = 1 → (𝑥 + 1) = (1 + 1))
2		oveq2 7398	. . 3 ⊢ (𝑥 = 1 → (1 + 𝑥) = (1 + 1))
3	1, 2	eqeq12d 2746	. 2 ⊢ (𝑥 = 1 → ((𝑥 + 1) = (1 + 𝑥) ↔ (1 + 1) = (1 + 1)))
4		oveq1 7397	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1))
5		oveq2 7398	. . 3 ⊢ (𝑥 = 𝑦 → (1 + 𝑥) = (1 + 𝑦))
6	4, 5	eqeq12d 2746	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 + 1) = (1 + 𝑥) ↔ (𝑦 + 1) = (1 + 𝑦)))
7		oveq1 7397	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1))
8		oveq2 7398	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (1 + 𝑥) = (1 + (𝑦 + 1)))
9	7, 8	eqeq12d 2746	. 2 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 + 1) = (1 + 𝑥) ↔ ((𝑦 + 1) + 1) = (1 + (𝑦 + 1))))
10		oveq1 7397	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 + 1) = (𝐴 + 1))
11		oveq2 7398	. . 3 ⊢ (𝑥 = 𝐴 → (1 + 𝑥) = (1 + 𝐴))
12	10, 11	eqeq12d 2746	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 + 1) = (1 + 𝑥) ↔ (𝐴 + 1) = (1 + 𝐴)))
13		eqid 2730	. 2 ⊢ (1 + 1) = (1 + 1)
14		oveq1 7397	. . . 4 ⊢ ((𝑦 + 1) = (1 + 𝑦) → ((𝑦 + 1) + 1) = ((1 + 𝑦) + 1))
15		1cnd 11176	. . . . 5 ⊢ (𝑦 ∈ ℕ → 1 ∈ ℂ)
16		nncn 12201	. . . . 5 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
17	15, 16, 15	addassd 11203	. . . 4 ⊢ (𝑦 ∈ ℕ → ((1 + 𝑦) + 1) = (1 + (𝑦 + 1)))
18	14, 17	sylan9eqr 2787	. . 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 12211	1 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) = (1 + 𝐴))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  (class class class)co 7390  1c1 11076   + caddc 11078  ℕcn 12193

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714  ax-1cn 11133  ax-addcl 11135  ax-addass 11140

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-nn 12194

This theorem is referenced by:  nnaddcom  42263  renegmulnnass  42460