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Theorem nnadd1com 42302
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 7438 . . 3 (𝑥 = 1 → (𝑥 + 1) = (1 + 1))
2 oveq2 7439 . . 3 (𝑥 = 1 → (1 + 𝑥) = (1 + 1))
31, 2eqeq12d 2753 . 2 (𝑥 = 1 → ((𝑥 + 1) = (1 + 𝑥) ↔ (1 + 1) = (1 + 1)))
4 oveq1 7438 . . 3 (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1))
5 oveq2 7439 . . 3 (𝑥 = 𝑦 → (1 + 𝑥) = (1 + 𝑦))
64, 5eqeq12d 2753 . 2 (𝑥 = 𝑦 → ((𝑥 + 1) = (1 + 𝑥) ↔ (𝑦 + 1) = (1 + 𝑦)))
7 oveq1 7438 . . 3 (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1))
8 oveq2 7439 . . 3 (𝑥 = (𝑦 + 1) → (1 + 𝑥) = (1 + (𝑦 + 1)))
97, 8eqeq12d 2753 . 2 (𝑥 = (𝑦 + 1) → ((𝑥 + 1) = (1 + 𝑥) ↔ ((𝑦 + 1) + 1) = (1 + (𝑦 + 1))))
10 oveq1 7438 . . 3 (𝑥 = 𝐴 → (𝑥 + 1) = (𝐴 + 1))
11 oveq2 7439 . . 3 (𝑥 = 𝐴 → (1 + 𝑥) = (1 + 𝐴))
1210, 11eqeq12d 2753 . 2 (𝑥 = 𝐴 → ((𝑥 + 1) = (1 + 𝑥) ↔ (𝐴 + 1) = (1 + 𝐴)))
13 eqid 2737 . 2 (1 + 1) = (1 + 1)
14 oveq1 7438 . . . 4 ((𝑦 + 1) = (1 + 𝑦) → ((𝑦 + 1) + 1) = ((1 + 𝑦) + 1))
15 1cnd 11256 . . . . 5 (𝑦 ∈ ℕ → 1 ∈ ℂ)
16 nncn 12274 . . . . 5 (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
1715, 16, 15addassd 11283 . . . 4 (𝑦 ∈ ℕ → ((1 + 𝑦) + 1) = (1 + (𝑦 + 1)))
1814, 17sylan9eqr 2799 . . 3 ((𝑦 ∈ ℕ ∧ (𝑦 + 1) = (1 + 𝑦)) → ((𝑦 + 1) + 1) = (1 + (𝑦 + 1)))
1918ex 412 . 2 (𝑦 ∈ ℕ → ((𝑦 + 1) = (1 + 𝑦) → ((𝑦 + 1) + 1) = (1 + (𝑦 + 1))))
203, 6, 9, 12, 13, 19nnind 12284 1 (𝐴 ∈ ℕ → (𝐴 + 1) = (1 + 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  (class class class)co 7431  1c1 11156   + caddc 11158  cn 12266
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 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-1cn 11213  ax-addcl 11215  ax-addass 11220
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-nn 12267
This theorem is referenced by:  nnaddcom  42303  renegmulnnass  42483
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