Theorem nnadd1com 39166

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 7166	. . 3 ⊢ (𝑥 = 1 → (𝑥 + 1) = (1 + 1))
2		oveq2 7167	. . 3 ⊢ (𝑥 = 1 → (1 + 𝑥) = (1 + 1))
3	1, 2	eqeq12d 2840	. 2 ⊢ (𝑥 = 1 → ((𝑥 + 1) = (1 + 𝑥) ↔ (1 + 1) = (1 + 1)))
4		oveq1 7166	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1))
5		oveq2 7167	. . 3 ⊢ (𝑥 = 𝑦 → (1 + 𝑥) = (1 + 𝑦))
6	4, 5	eqeq12d 2840	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 + 1) = (1 + 𝑥) ↔ (𝑦 + 1) = (1 + 𝑦)))
7		oveq1 7166	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1))
8		oveq2 7167	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (1 + 𝑥) = (1 + (𝑦 + 1)))
9	7, 8	eqeq12d 2840	. 2 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 + 1) = (1 + 𝑥) ↔ ((𝑦 + 1) + 1) = (1 + (𝑦 + 1))))
10		oveq1 7166	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 + 1) = (𝐴 + 1))
11		oveq2 7167	. . 3 ⊢ (𝑥 = 𝐴 → (1 + 𝑥) = (1 + 𝐴))
12	10, 11	eqeq12d 2840	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 + 1) = (1 + 𝑥) ↔ (𝐴 + 1) = (1 + 𝐴)))
13		eqid 2824	. 2 ⊢ (1 + 1) = (1 + 1)
14		oveq1 7166	. . . 4 ⊢ ((𝑦 + 1) = (1 + 𝑦) → ((𝑦 + 1) + 1) = ((1 + 𝑦) + 1))
15		1cnd 10639	. . . . 5 ⊢ (𝑦 ∈ ℕ → 1 ∈ ℂ)
16		nncn 11649	. . . . 5 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
17	15, 16, 15	addassd 10666	. . . 4 ⊢ (𝑦 ∈ ℕ → ((1 + 𝑦) + 1) = (1 + (𝑦 + 1)))
18	14, 17	sylan9eqr 2881	. . 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 11659	1 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) = (1 + 𝐴))

Syntax hints:   → wi 4   = wceq 1536   ∈ wcel 2113  (class class class)co 7159  1c1 10541   + caddc 10543  ℕcn 11641

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-1cn 10598  ax-addcl 10600  ax-addass 10605

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-om 7584  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-nn 11642

This theorem is referenced by:  nnaddcom  39167