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Theorem nnadd1com 42281
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 2751 . 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 2751 . 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 2751 . 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 2751 . 2 (𝑥 = 𝐴 → ((𝑥 + 1) = (1 + 𝑥) ↔ (𝐴 + 1) = (1 + 𝐴)))
13 eqid 2735 . 2 (1 + 1) = (1 + 1)
14 oveq1 7438 . . . 4 ((𝑦 + 1) = (1 + 𝑦) → ((𝑦 + 1) + 1) = ((1 + 𝑦) + 1))
15 1cnd 11254 . . . . 5 (𝑦 ∈ ℕ → 1 ∈ ℂ)
16 nncn 12272 . . . . 5 (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
1715, 16, 15addassd 11281 . . . 4 (𝑦 ∈ ℕ → ((1 + 𝑦) + 1) = (1 + (𝑦 + 1)))
1814, 17sylan9eqr 2797 . . 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 12282 1 (𝐴 ∈ ℕ → (𝐴 + 1) = (1 + 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  (class class class)co 7431  1c1 11154   + caddc 11156  cn 12264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-1cn 11211  ax-addcl 11213  ax-addass 11218
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-nn 12265
This theorem is referenced by:  nnaddcom  42282  renegmulnnass  42460
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