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Theorem nnadd1com 39511
 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.
StepHypRef Expression
1 oveq1 7143 . . 3 (𝑥 = 1 → (𝑥 + 1) = (1 + 1))
2 oveq2 7144 . . 3 (𝑥 = 1 → (1 + 𝑥) = (1 + 1))
31, 2eqeq12d 2814 . 2 (𝑥 = 1 → ((𝑥 + 1) = (1 + 𝑥) ↔ (1 + 1) = (1 + 1)))
4 oveq1 7143 . . 3 (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1))
5 oveq2 7144 . . 3 (𝑥 = 𝑦 → (1 + 𝑥) = (1 + 𝑦))
64, 5eqeq12d 2814 . 2 (𝑥 = 𝑦 → ((𝑥 + 1) = (1 + 𝑥) ↔ (𝑦 + 1) = (1 + 𝑦)))
7 oveq1 7143 . . 3 (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1))
8 oveq2 7144 . . 3 (𝑥 = (𝑦 + 1) → (1 + 𝑥) = (1 + (𝑦 + 1)))
97, 8eqeq12d 2814 . 2 (𝑥 = (𝑦 + 1) → ((𝑥 + 1) = (1 + 𝑥) ↔ ((𝑦 + 1) + 1) = (1 + (𝑦 + 1))))
10 oveq1 7143 . . 3 (𝑥 = 𝐴 → (𝑥 + 1) = (𝐴 + 1))
11 oveq2 7144 . . 3 (𝑥 = 𝐴 → (1 + 𝑥) = (1 + 𝐴))
1210, 11eqeq12d 2814 . 2 (𝑥 = 𝐴 → ((𝑥 + 1) = (1 + 𝑥) ↔ (𝐴 + 1) = (1 + 𝐴)))
13 eqid 2798 . 2 (1 + 1) = (1 + 1)
14 oveq1 7143 . . . 4 ((𝑦 + 1) = (1 + 𝑦) → ((𝑦 + 1) + 1) = ((1 + 𝑦) + 1))
15 1cnd 10628 . . . . 5 (𝑦 ∈ ℕ → 1 ∈ ℂ)
16 nncn 11636 . . . . 5 (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
1715, 16, 15addassd 10655 . . . 4 (𝑦 ∈ ℕ → ((1 + 𝑦) + 1) = (1 + (𝑦 + 1)))
1814, 17sylan9eqr 2855 . . 3 ((𝑦 ∈ ℕ ∧ (𝑦 + 1) = (1 + 𝑦)) → ((𝑦 + 1) + 1) = (1 + (𝑦 + 1)))
1918ex 416 . 2 (𝑦 ∈ ℕ → ((𝑦 + 1) = (1 + 𝑦) → ((𝑦 + 1) + 1) = (1 + (𝑦 + 1))))
203, 6, 9, 12, 13, 19nnind 11646 1 (𝐴 ∈ ℕ → (𝐴 + 1) = (1 + 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  (class class class)co 7136  1c1 10530   + caddc 10532  ℕcn 11628 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-1cn 10587  ax-addcl 10589  ax-addass 10594 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-om 7564  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-nn 11629 This theorem is referenced by:  nnaddcom  39512
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