Theorem nnmcom 3183

Description: Multiplication of natural numbers is commutative. Theorem 4K(5) of [Enderton] p. 81.

Assertion

Ref	Expression
nnmcom	⊢ ((A ∈ ω ∧ B ∈ ω) → (A ·o B) = (B ·o A))

Proof of Theorem nnmcom

Step	Hyp	Ref	Expression
1		opreq1 3006	. . . . 5 ⊢ (x = ∅ → (x ·o B) = (∅ ·o B))
2		opreq2 3007	. . . . 5 ⊢ (x = ∅ → (B ·o x) = (B ·o ∅))
3	1, 2	cleq12d 1115	. . . 4 ⊢ (x = ∅ → ((x ·o B) = (B ·o x) ↔ (∅ ·o B) = (B ·o ∅)))
4	3	imbi2d 464	. . 3 ⊢ (x = ∅ → ((B ∈ ω → (x ·o B) = (B ·o x)) ↔ (B ∈ ω → (∅ ·o B) = (B ·o ∅))))
5		opreq1 3006	. . . . 5 ⊢ (x = y → (x ·o B) = (y ·o B))
6		opreq2 3007	. . . . 5 ⊢ (x = y → (B ·o x) = (B ·o y))
7	5, 6	cleq12d 1115	. . . 4 ⊢ (x = y → ((x ·o B) = (B ·o x) ↔ (y ·o B) = (B ·o y)))
8	7	imbi2d 464	. . 3 ⊢ (x = y → ((B ∈ ω → (x ·o B) = (B ·o x)) ↔ (B ∈ ω → (y ·o B) = (B ·o y))))
9		opreq1 3006	. . . . 5 ⊢ (x = suc y → (x ·o B) = (suc y ·o B))
10		opreq2 3007	. . . . 5 ⊢ (x = suc y → (B ·o x) = (B ·o suc y))
11	9, 10	cleq12d 1115	. . . 4 ⊢ (x = suc y → ((x ·o B) = (B ·o x) ↔ (suc y ·o B) = (B ·o suc y)))
12	11	imbi2d 464	. . 3 ⊢ (x = suc y → ((B ∈ ω → (x ·o B) = (B ·o x)) ↔ (B ∈ ω → (suc y ·o B) = (B ·o suc y))))
13		opreq1 3006	. . . . 5 ⊢ (x = A → (x ·o B) = (A ·o B))
14		opreq2 3007	. . . . 5 ⊢ (x = A → (B ·o x) = (B ·o A))
15	13, 14	cleq12d 1115	. . . 4 ⊢ (x = A → ((x ·o B) = (B ·o x) ↔ (A ·o B) = (B ·o A)))
16	15	imbi2d 464	. . 3 ⊢ (x = A → ((B ∈ ω → (x ·o B) = (B ·o x)) ↔ (B ∈ ω → (A ·o B) = (B ·o A))))
17		nnm0r 3171	. . . 4 ⊢ (B ∈ ω → (∅ ·o B) = ∅)
18		nnm0 3167	. . . 4 ⊢ (B ∈ ω → (B ·o ∅) = ∅)
19	17, 18	eqtr4d 1131	. . 3 ⊢ (B ∈ ω → (∅ ·o B) = (B ·o ∅))
20		nnmsucr 3182	. . . . . . 7 ⊢ ((y ∈ ω ∧ B ∈ ω) → (suc y ·o B) = ((y ·o B) +o B))
21		nnmsuc 3169	. . . . . . . 8 ⊢ ((B ∈ ω ∧ y ∈ ω) → (B ·o suc y) = ((B ·o y) +o B))
22	21	ancoms 334	. . . . . . 7 ⊢ ((y ∈ ω ∧ B ∈ ω) → (B ·o suc y) = ((B ·o y) +o B))
23	20, 22	cleq12d 1115	. . . . . 6 ⊢ ((y ∈ ω ∧ B ∈ ω) → ((suc y ·o B) = (B ·o suc y) ↔ ((y ·o B) +o B) = ((B ·o y) +o B)))
24		opreq1 3006	. . . . . 6 ⊢ ((y ·o B) = (B ·o y) → ((y ·o B) +o B) = ((B ·o y) +o B))
25	23, 24	syl5bir 184	. . . . 5 ⊢ ((y ∈ ω ∧ B ∈ ω) → ((y ·o B) = (B ·o y) → (suc y ·o B) = (B ·o suc y)))
26	25	exp 291	. . . 4 ⊢ (y ∈ ω → (B ∈ ω → ((y ·o B) = (B ·o y) → (suc y ·o B) = (B ·o suc y))))
27	26	a2d 15	. . 3 ⊢ (y ∈ ω → ((B ∈ ω → (y ·o B) = (B ·o y)) → (B ∈ ω → (suc y ·o B) = (B ·o suc y))))
28	4, 8, 12, 16, 19, 27	finds 2397	. 2 ⊢ (A ∈ ω → (B ∈ ω → (A ·o B) = (B ·o A)))
29	28	imp 277	1 ⊢ ((A ∈ ω ∧ B ∈ ω) → (A ·o B) = (B ·o A))

Syntax hints:   → wi 2   ∧ wa 196   = weq 797   = wceq 1091   ∈ wcel 1092  ∅c0 1707  suc csuc 2201  ωcom 2372  (class class class)co 3001   +_o coa 3101   ·_o comu 3102

This theorem is referenced by:  mulcompi 3818

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-oadd 3106  df-omul 3107

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