Users' Mathboxes Mathbox for Steven Nguyen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nnaddcom Structured version   Visualization version   GIF version

Theorem nnaddcom 41132
Description: Addition is commutative for natural numbers. Uses fewer axioms than addcom 11396. (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 7411 . . . . 5 (𝑥 = 1 → (𝑥 + 𝐵) = (1 + 𝐵))
2 oveq2 7412 . . . . 5 (𝑥 = 1 → (𝐵 + 𝑥) = (𝐵 + 1))
31, 2eqeq12d 2749 . . . 4 (𝑥 = 1 → ((𝑥 + 𝐵) = (𝐵 + 𝑥) ↔ (1 + 𝐵) = (𝐵 + 1)))
43imbi2d 341 . . 3 (𝑥 = 1 → ((𝐵 ∈ ℕ → (𝑥 + 𝐵) = (𝐵 + 𝑥)) ↔ (𝐵 ∈ ℕ → (1 + 𝐵) = (𝐵 + 1))))
5 oveq1 7411 . . . . 5 (𝑥 = 𝑦 → (𝑥 + 𝐵) = (𝑦 + 𝐵))
6 oveq2 7412 . . . . 5 (𝑥 = 𝑦 → (𝐵 + 𝑥) = (𝐵 + 𝑦))
75, 6eqeq12d 2749 . . . 4 (𝑥 = 𝑦 → ((𝑥 + 𝐵) = (𝐵 + 𝑥) ↔ (𝑦 + 𝐵) = (𝐵 + 𝑦)))
87imbi2d 341 . . 3 (𝑥 = 𝑦 → ((𝐵 ∈ ℕ → (𝑥 + 𝐵) = (𝐵 + 𝑥)) ↔ (𝐵 ∈ ℕ → (𝑦 + 𝐵) = (𝐵 + 𝑦))))
9 oveq1 7411 . . . . 5 (𝑥 = (𝑦 + 1) → (𝑥 + 𝐵) = ((𝑦 + 1) + 𝐵))
10 oveq2 7412 . . . . 5 (𝑥 = (𝑦 + 1) → (𝐵 + 𝑥) = (𝐵 + (𝑦 + 1)))
119, 10eqeq12d 2749 . . . 4 (𝑥 = (𝑦 + 1) → ((𝑥 + 𝐵) = (𝐵 + 𝑥) ↔ ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1))))
1211imbi2d 341 . . 3 (𝑥 = (𝑦 + 1) → ((𝐵 ∈ ℕ → (𝑥 + 𝐵) = (𝐵 + 𝑥)) ↔ (𝐵 ∈ ℕ → ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1)))))
13 oveq1 7411 . . . . 5 (𝑥 = 𝐴 → (𝑥 + 𝐵) = (𝐴 + 𝐵))
14 oveq2 7412 . . . . 5 (𝑥 = 𝐴 → (𝐵 + 𝑥) = (𝐵 + 𝐴))
1513, 14eqeq12d 2749 . . . 4 (𝑥 = 𝐴 → ((𝑥 + 𝐵) = (𝐵 + 𝑥) ↔ (𝐴 + 𝐵) = (𝐵 + 𝐴)))
1615imbi2d 341 . . 3 (𝑥 = 𝐴 → ((𝐵 ∈ ℕ → (𝑥 + 𝐵) = (𝐵 + 𝑥)) ↔ (𝐵 ∈ ℕ → (𝐴 + 𝐵) = (𝐵 + 𝐴))))
17 nnadd1com 41131 . . . 4 (𝐵 ∈ ℕ → (𝐵 + 1) = (1 + 𝐵))
1817eqcomd 2739 . . 3 (𝐵 ∈ ℕ → (1 + 𝐵) = (𝐵 + 1))
19 oveq1 7411 . . . . . 6 ((𝑦 + 𝐵) = (𝐵 + 𝑦) → ((𝑦 + 𝐵) + 1) = ((𝐵 + 𝑦) + 1))
2017oveq2d 7420 . . . . . . . . 9 (𝐵 ∈ ℕ → (𝑦 + (𝐵 + 1)) = (𝑦 + (1 + 𝐵)))
2120adantl 483 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑦 + (𝐵 + 1)) = (𝑦 + (1 + 𝐵)))
22 nncn 12216 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
2322adantr 482 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝑦 ∈ ℂ)
24 nncn 12216 . . . . . . . . . 10 (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
2524adantl 483 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℂ)
26 1cnd 11205 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 1 ∈ ℂ)
2723, 25, 26addassd 11232 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝑦 + 𝐵) + 1) = (𝑦 + (𝐵 + 1)))
2823, 26, 25addassd 11232 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝑦 + 1) + 𝐵) = (𝑦 + (1 + 𝐵)))
2921, 27, 283eqtr4d 2783 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝑦 + 𝐵) + 1) = ((𝑦 + 1) + 𝐵))
3025, 23, 26addassd 11232 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐵 + 𝑦) + 1) = (𝐵 + (𝑦 + 1)))
3129, 30eqeq12d 2749 . . . . . 6 ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (((𝑦 + 𝐵) + 1) = ((𝐵 + 𝑦) + 1) ↔ ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1))))
3219, 31imbitrid 243 . . . . 5 ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝑦 + 𝐵) = (𝐵 + 𝑦) → ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1))))
3332ex 414 . . . 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 12226 . 2 (𝐴 ∈ ℕ → (𝐵 ∈ ℕ → (𝐴 + 𝐵) = (𝐵 + 𝐴)))
3635imp 408 1 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  (class class class)co 7404  cc 11104  1c1 11107   + caddc 11109  cn 12208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720  ax-1cn 11164  ax-addcl 11166  ax-addass 11171
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-om 7851  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-nn 12209
This theorem is referenced by:  nnaddcomli  41133  nnadddir  41134  nn0addcom  41267  zaddcom  41269
  Copyright terms: Public domain W3C validator