Theorem nnaddcom 39210

Description: Addition is commutative for natural numbers. Uses fewer axioms than addcom 10826. (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.

Step	Hyp	Ref	Expression
1		oveq1 7163	. . . . 5 ⊢ (𝑥 = 1 → (𝑥 + 𝐵) = (1 + 𝐵))
2		oveq2 7164	. . . . 5 ⊢ (𝑥 = 1 → (𝐵 + 𝑥) = (𝐵 + 1))
3	1, 2	eqeq12d 2837	. . . 4 ⊢ (𝑥 = 1 → ((𝑥 + 𝐵) = (𝐵 + 𝑥) ↔ (1 + 𝐵) = (𝐵 + 1)))
4	3	imbi2d 343	. . 3 ⊢ (𝑥 = 1 → ((𝐵 ∈ ℕ → (𝑥 + 𝐵) = (𝐵 + 𝑥)) ↔ (𝐵 ∈ ℕ → (1 + 𝐵) = (𝐵 + 1))))
5		oveq1 7163	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 + 𝐵) = (𝑦 + 𝐵))
6		oveq2 7164	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 + 𝑥) = (𝐵 + 𝑦))
7	5, 6	eqeq12d 2837	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 + 𝐵) = (𝐵 + 𝑥) ↔ (𝑦 + 𝐵) = (𝐵 + 𝑦)))
8	7	imbi2d 343	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐵 ∈ ℕ → (𝑥 + 𝐵) = (𝐵 + 𝑥)) ↔ (𝐵 ∈ ℕ → (𝑦 + 𝐵) = (𝐵 + 𝑦))))
9		oveq1 7163	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 + 𝐵) = ((𝑦 + 1) + 𝐵))
10		oveq2 7164	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝐵 + 𝑥) = (𝐵 + (𝑦 + 1)))
11	9, 10	eqeq12d 2837	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 + 𝐵) = (𝐵 + 𝑥) ↔ ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1))))
12	11	imbi2d 343	. . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝐵 ∈ ℕ → (𝑥 + 𝐵) = (𝐵 + 𝑥)) ↔ (𝐵 ∈ ℕ → ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1)))))
13		oveq1 7163	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 + 𝐵) = (𝐴 + 𝐵))
14		oveq2 7164	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 + 𝑥) = (𝐵 + 𝐴))
15	13, 14	eqeq12d 2837	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 + 𝐵) = (𝐵 + 𝑥) ↔ (𝐴 + 𝐵) = (𝐵 + 𝐴)))
16	15	imbi2d 343	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ ℕ → (𝑥 + 𝐵) = (𝐵 + 𝑥)) ↔ (𝐵 ∈ ℕ → (𝐴 + 𝐵) = (𝐵 + 𝐴))))
17		nnadd1com 39209	. . . 4 ⊢ (𝐵 ∈ ℕ → (𝐵 + 1) = (1 + 𝐵))
18	17	eqcomd 2827	. . 3 ⊢ (𝐵 ∈ ℕ → (1 + 𝐵) = (𝐵 + 1))
19		oveq1 7163	. . . . . 6 ⊢ ((𝑦 + 𝐵) = (𝐵 + 𝑦) → ((𝑦 + 𝐵) + 1) = ((𝐵 + 𝑦) + 1))
20	17	oveq2d 7172	. . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → (𝑦 + (𝐵 + 1)) = (𝑦 + (1 + 𝐵)))
21	20	adantl 484	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑦 + (𝐵 + 1)) = (𝑦 + (1 + 𝐵)))
22		nncn 11646	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
23	22	adantr 483	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝑦 ∈ ℂ)
24		nncn 11646	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
25	24	adantl 484	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℂ)
26		1cnd 10636	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 1 ∈ ℂ)
27	23, 25, 26	addassd 10663	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝑦 + 𝐵) + 1) = (𝑦 + (𝐵 + 1)))
28	23, 26, 25	addassd 10663	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝑦 + 1) + 𝐵) = (𝑦 + (1 + 𝐵)))
29	21, 27, 28	3eqtr4d 2866	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝑦 + 𝐵) + 1) = ((𝑦 + 1) + 𝐵))
30	25, 23, 26	addassd 10663	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐵 + 𝑦) + 1) = (𝐵 + (𝑦 + 1)))
31	29, 30	eqeq12d 2837	. . . . . 6 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (((𝑦 + 𝐵) + 1) = ((𝐵 + 𝑦) + 1) ↔ ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1))))
32	19, 31	syl5ib 246	. . . . 5 ⊢ ((𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝑦 + 𝐵) = (𝐵 + 𝑦) → ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1))))
33	32	ex 415	. . . 4 ⊢ (𝑦 ∈ ℕ → (𝐵 ∈ ℕ → ((𝑦 + 𝐵) = (𝐵 + 𝑦) → ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1)))))
34	33	a2d 29	. . 3 ⊢ (𝑦 ∈ ℕ → ((𝐵 ∈ ℕ → (𝑦 + 𝐵) = (𝐵 + 𝑦)) → (𝐵 ∈ ℕ → ((𝑦 + 1) + 𝐵) = (𝐵 + (𝑦 + 1)))))
35	4, 8, 12, 16, 18, 34	nnind 11656	. 2 ⊢ (𝐴 ∈ ℕ → (𝐵 ∈ ℕ → (𝐴 + 𝐵) = (𝐵 + 𝐴)))
36	35	imp 409	1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  (class class class)co 7156  ℂcc 10535  1c1 10538   + caddc 10540  ℕcn 11638

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-1cn 10595  ax-addcl 10597  ax-addass 10602

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-om 7581  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-nn 11639

This theorem is referenced by:  nnaddcomli  39211  nnadddir  39212