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Theorem nnadd1com 41484
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 7419 . . 3 (𝑥 = 1 → (𝑥 + 1) = (1 + 1))
2 oveq2 7420 . . 3 (𝑥 = 1 → (1 + 𝑥) = (1 + 1))
31, 2eqeq12d 2747 . 2 (𝑥 = 1 → ((𝑥 + 1) = (1 + 𝑥) ↔ (1 + 1) = (1 + 1)))
4 oveq1 7419 . . 3 (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1))
5 oveq2 7420 . . 3 (𝑥 = 𝑦 → (1 + 𝑥) = (1 + 𝑦))
64, 5eqeq12d 2747 . 2 (𝑥 = 𝑦 → ((𝑥 + 1) = (1 + 𝑥) ↔ (𝑦 + 1) = (1 + 𝑦)))
7 oveq1 7419 . . 3 (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1))
8 oveq2 7420 . . 3 (𝑥 = (𝑦 + 1) → (1 + 𝑥) = (1 + (𝑦 + 1)))
97, 8eqeq12d 2747 . 2 (𝑥 = (𝑦 + 1) → ((𝑥 + 1) = (1 + 𝑥) ↔ ((𝑦 + 1) + 1) = (1 + (𝑦 + 1))))
10 oveq1 7419 . . 3 (𝑥 = 𝐴 → (𝑥 + 1) = (𝐴 + 1))
11 oveq2 7420 . . 3 (𝑥 = 𝐴 → (1 + 𝑥) = (1 + 𝐴))
1210, 11eqeq12d 2747 . 2 (𝑥 = 𝐴 → ((𝑥 + 1) = (1 + 𝑥) ↔ (𝐴 + 1) = (1 + 𝐴)))
13 eqid 2731 . 2 (1 + 1) = (1 + 1)
14 oveq1 7419 . . . 4 ((𝑦 + 1) = (1 + 𝑦) → ((𝑦 + 1) + 1) = ((1 + 𝑦) + 1))
15 1cnd 11214 . . . . 5 (𝑦 ∈ ℕ → 1 ∈ ℂ)
16 nncn 12225 . . . . 5 (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
1715, 16, 15addassd 11241 . . . 4 (𝑦 ∈ ℕ → ((1 + 𝑦) + 1) = (1 + (𝑦 + 1)))
1814, 17sylan9eqr 2793 . . 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 12235 1 (𝐴 ∈ ℕ → (𝐴 + 1) = (1 + 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  (class class class)co 7412  1c1 11115   + caddc 11117  cn 12217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729  ax-1cn 11172  ax-addcl 11174  ax-addass 11179
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-om 7860  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-nn 12218
This theorem is referenced by:  nnaddcom  41485  renegmulnnass  41629
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