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Theorem nnaddcom 42282
Description: Addition is commutative for natural numbers. Uses fewer axioms than addcom 11445. (Contributed by Steven Nguyen, 9-Dec-2022.)
Assertion
Ref Expression
nnaddcom ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Proof of Theorem nnaddcom
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7438 . . . . 5 (𝑥 = 1 → (𝑥 + 𝐵) = (1 + 𝐵))
2 oveq2 7439 . . . . 5 (𝑥 = 1 → (𝐵 + 𝑥) = (𝐵 + 1))
31, 2eqeq12d 2751 . . . 4 (𝑥 = 1 → ((𝑥 + 𝐵) = (𝐵 + 𝑥) ↔ (1 + 𝐵) = (𝐵 + 1)))
43imbi2d 340 . . 3 (𝑥 = 1 → ((𝐵 ∈ ℕ → (𝑥 + 𝐵) = (𝐵 + 𝑥)) ↔ (𝐵 ∈ ℕ → (1 + 𝐵) = (𝐵 + 1))))
5 oveq1 7438 . . . . 5 (𝑥 = 𝑦 → (𝑥 + 𝐵) = (𝑦 + 𝐵))
6 oveq2 7439 . . . . 5 (𝑥 = 𝑦 → (𝐵 + 𝑥) = (𝐵 + 𝑦))
75, 6eqeq12d 2751 . . . 4 (𝑥 = 𝑦 → ((𝑥 + 𝐵) = (𝐵 + 𝑥) ↔ (𝑦 + 𝐵) = (𝐵 + 𝑦)))
87imbi2d 340 . . 3 (𝑥 = 𝑦 → ((𝐵 ∈ ℕ → (𝑥 + 𝐵) = (𝐵 + 𝑥)) ↔ (𝐵 ∈ ℕ → (𝑦 + 𝐵) = (𝐵 + 𝑦))))
9 oveq1 7438 . . . . 5 (𝑥 = (𝑦 + 1) → (𝑥 + 𝐵) = ((𝑦 + 1) + 𝐵))
10 oveq2 7439 . . . . 5 (𝑥 = (𝑦 + 1) → (𝐵 + 𝑥) = (𝐵 + (𝑦 + 1)))
119, 10eqeq12d 2751 . . . 4 (𝑥 = (𝑦 + 1) → ((𝑥 + 𝐵) = (𝐵 + 𝑥) ↔ ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1))))
1211imbi2d 340 . . 3 (𝑥 = (𝑦 + 1) → ((𝐵 ∈ ℕ → (𝑥 + 𝐵) = (𝐵 + 𝑥)) ↔ (𝐵 ∈ ℕ → ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1)))))
13 oveq1 7438 . . . . 5 (𝑥 = 𝐴 → (𝑥 + 𝐵) = (𝐴 + 𝐵))
14 oveq2 7439 . . . . 5 (𝑥 = 𝐴 → (𝐵 + 𝑥) = (𝐵 + 𝐴))
1513, 14eqeq12d 2751 . . . 4 (𝑥 = 𝐴 → ((𝑥 + 𝐵) = (𝐵 + 𝑥) ↔ (𝐴 + 𝐵) = (𝐵 + 𝐴)))
1615imbi2d 340 . . 3 (𝑥 = 𝐴 → ((𝐵 ∈ ℕ → (𝑥 + 𝐵) = (𝐵 + 𝑥)) ↔ (𝐵 ∈ ℕ → (𝐴 + 𝐵) = (𝐵 + 𝐴))))
17 nnadd1com 42281 . . . 4 (𝐵 ∈ ℕ → (𝐵 + 1) = (1 + 𝐵))
1817eqcomd 2741 . . 3 (𝐵 ∈ ℕ → (1 + 𝐵) = (𝐵 + 1))
19 oveq1 7438 . . . . . 6 ((𝑦 + 𝐵) = (𝐵 + 𝑦) → ((𝑦 + 𝐵) + 1) = ((𝐵 + 𝑦) + 1))
2017oveq2d 7447 . . . . . . . . 9 (𝐵 ∈ ℕ → (𝑦 + (𝐵 + 1)) = (𝑦 + (1 + 𝐵)))
2120adantl 481 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑦 + (𝐵 + 1)) = (𝑦 + (1 + 𝐵)))
22 nncn 12272 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
2322adantr 480 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝑦 ∈ ℂ)
24 nncn 12272 . . . . . . . . . 10 (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
2524adantl 481 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℂ)
26 1cnd 11254 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 1 ∈ ℂ)
2723, 25, 26addassd 11281 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝑦 + 𝐵) + 1) = (𝑦 + (𝐵 + 1)))
2823, 26, 25addassd 11281 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝑦 + 1) + 𝐵) = (𝑦 + (1 + 𝐵)))
2921, 27, 283eqtr4d 2785 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝑦 + 𝐵) + 1) = ((𝑦 + 1) + 𝐵))
3025, 23, 26addassd 11281 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐵 + 𝑦) + 1) = (𝐵 + (𝑦 + 1)))
3129, 30eqeq12d 2751 . . . . . 6 ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (((𝑦 + 𝐵) + 1) = ((𝐵 + 𝑦) + 1) ↔ ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1))))
3219, 31imbitrid 244 . . . . 5 ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝑦 + 𝐵) = (𝐵 + 𝑦) → ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1))))
3332ex 412 . . . 4 (𝑦 ∈ ℕ → (𝐵 ∈ ℕ → ((𝑦 + 𝐵) = (𝐵 + 𝑦) → ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1)))))
3433a2d 29 . . 3 (𝑦 ∈ ℕ → ((𝐵 ∈ ℕ → (𝑦 + 𝐵) = (𝐵 + 𝑦)) → (𝐵 ∈ ℕ → ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1)))))
354, 8, 12, 16, 18, 34nnind 12282 . 2 (𝐴 ∈ ℕ → (𝐵 ∈ ℕ → (𝐴 + 𝐵) = (𝐵 + 𝐴)))
3635imp 406 1 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  (class class class)co 7431  cc 11151  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:  nnaddcomli  42283  nnadddir  42284  nn0addcom  42457  zaddcom  42459
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