Metamath Proof Explorer

Theorem nnmass

Description: Multiplication of natural numbers is associative. Theorem 4K(4) of Enderton p. 81. (Contributed by NM, 20-Sep-1995) (Revised by Mario Carneiro, 15-Nov-2014)

Ref Expression
Assertion nnmass ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) → ( ( 𝐴 ·o 𝐵 ) ·o 𝐶 ) = ( 𝐴 ·o ( 𝐵 ·o 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 oveq2 ⊢ ( 𝑥 = 𝐶 → ( ( 𝐴 ·o 𝐵 ) ·o 𝑥 ) = ( ( 𝐴 ·o 𝐵 ) ·o 𝐶 ) )
2 oveq2 ⊢ ( 𝑥 = 𝐶 → ( 𝐵 ·o 𝑥 ) = ( 𝐵 ·o 𝐶 ) )
3 2 oveq2d ⊢ ( 𝑥 = 𝐶 → ( 𝐴 ·o ( 𝐵 ·o 𝑥 ) ) = ( 𝐴 ·o ( 𝐵 ·o 𝐶 ) ) )
4 1 3 eqeq12d ⊢ ( 𝑥 = 𝐶 → ( ( ( 𝐴 ·o 𝐵 ) ·o 𝑥 ) = ( 𝐴 ·o ( 𝐵 ·o 𝑥 ) ) ↔ ( ( 𝐴 ·o 𝐵 ) ·o 𝐶 ) = ( 𝐴 ·o ( 𝐵 ·o 𝐶 ) ) ) )
5 4 imbi2d ⊢ ( 𝑥 = 𝐶 → ( ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( ( 𝐴 ·o 𝐵 ) ·o 𝑥 ) = ( 𝐴 ·o ( 𝐵 ·o 𝑥 ) ) ) ↔ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( ( 𝐴 ·o 𝐵 ) ·o 𝐶 ) = ( 𝐴 ·o ( 𝐵 ·o 𝐶 ) ) ) ) )
6 oveq2 ⊢ ( 𝑥 = ∅ → ( ( 𝐴 ·o 𝐵 ) ·o 𝑥 ) = ( ( 𝐴 ·o 𝐵 ) ·o ∅ ) )
7 oveq2 ⊢ ( 𝑥 = ∅ → ( 𝐵 ·o 𝑥 ) = ( 𝐵 ·o ∅ ) )
8 7 oveq2d ⊢ ( 𝑥 = ∅ → ( 𝐴 ·o ( 𝐵 ·o 𝑥 ) ) = ( 𝐴 ·o ( 𝐵 ·o ∅ ) ) )
9 6 8 eqeq12d ⊢ ( 𝑥 = ∅ → ( ( ( 𝐴 ·o 𝐵 ) ·o 𝑥 ) = ( 𝐴 ·o ( 𝐵 ·o 𝑥 ) ) ↔ ( ( 𝐴 ·o 𝐵 ) ·o ∅ ) = ( 𝐴 ·o ( 𝐵 ·o ∅ ) ) ) )
10 oveq2 ⊢ ( 𝑥 = 𝑦 → ( ( 𝐴 ·o 𝐵 ) ·o 𝑥 ) = ( ( 𝐴 ·o 𝐵 ) ·o 𝑦 ) )
11 oveq2 ⊢ ( 𝑥 = 𝑦 → ( 𝐵 ·o 𝑥 ) = ( 𝐵 ·o 𝑦 ) )
12 11 oveq2d ⊢ ( 𝑥 = 𝑦 → ( 𝐴 ·o ( 𝐵 ·o 𝑥 ) ) = ( 𝐴 ·o ( 𝐵 ·o 𝑦 ) ) )
13 10 12 eqeq12d ⊢ ( 𝑥 = 𝑦 → ( ( ( 𝐴 ·o 𝐵 ) ·o 𝑥 ) = ( 𝐴 ·o ( 𝐵 ·o 𝑥 ) ) ↔ ( ( 𝐴 ·o 𝐵 ) ·o 𝑦 ) = ( 𝐴 ·o ( 𝐵 ·o 𝑦 ) ) ) )
14 oveq2 ⊢ ( 𝑥 = suc 𝑦 → ( ( 𝐴 ·o 𝐵 ) ·o 𝑥 ) = ( ( 𝐴 ·o 𝐵 ) ·o suc 𝑦 ) )
15 oveq2 ⊢ ( 𝑥 = suc 𝑦 → ( 𝐵 ·o 𝑥 ) = ( 𝐵 ·o suc 𝑦 ) )
16 15 oveq2d ⊢ ( 𝑥 = suc 𝑦 → ( 𝐴 ·o ( 𝐵 ·o 𝑥 ) ) = ( 𝐴 ·o ( 𝐵 ·o suc 𝑦 ) ) )
17 14 16 eqeq12d ⊢ ( 𝑥 = suc 𝑦 → ( ( ( 𝐴 ·o 𝐵 ) ·o 𝑥 ) = ( 𝐴 ·o ( 𝐵 ·o 𝑥 ) ) ↔ ( ( 𝐴 ·o 𝐵 ) ·o suc 𝑦 ) = ( 𝐴 ·o ( 𝐵 ·o suc 𝑦 ) ) ) )
18 nnmcl ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ·o 𝐵 ) ∈ ω )
19 nnm0 ⊢ ( ( 𝐴 ·o 𝐵 ) ∈ ω → ( ( 𝐴 ·o 𝐵 ) ·o ∅ ) = ∅ )
20 18 19 syl ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( ( 𝐴 ·o 𝐵 ) ·o ∅ ) = ∅ )
21 nnm0 ⊢ ( 𝐵 ∈ ω → ( 𝐵 ·o ∅ ) = ∅ )
22 21 oveq2d ⊢ ( 𝐵 ∈ ω → ( 𝐴 ·o ( 𝐵 ·o ∅ ) ) = ( 𝐴 ·o ∅ ) )
23 nnm0 ⊢ ( 𝐴 ∈ ω → ( 𝐴 ·o ∅ ) = ∅ )
24 22 23 sylan9eqr ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ·o ( 𝐵 ·o ∅ ) ) = ∅ )
25 20 24 eqtr4d ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( ( 𝐴 ·o 𝐵 ) ·o ∅ ) = ( 𝐴 ·o ( 𝐵 ·o ∅ ) ) )
26 oveq1 ⊢ ( ( ( 𝐴 ·o 𝐵 ) ·o 𝑦 ) = ( 𝐴 ·o ( 𝐵 ·o 𝑦 ) ) → ( ( ( 𝐴 ·o 𝐵 ) ·o 𝑦 ) +o ( 𝐴 ·o 𝐵 ) ) = ( ( 𝐴 ·o ( 𝐵 ·o 𝑦 ) ) +o ( 𝐴 ·o 𝐵 ) ) )
27 nnmsuc ⊢ ( ( ( 𝐴 ·o 𝐵 ) ∈ ω ∧ 𝑦 ∈ ω ) → ( ( 𝐴 ·o 𝐵 ) ·o suc 𝑦 ) = ( ( ( 𝐴 ·o 𝐵 ) ·o 𝑦 ) +o ( 𝐴 ·o 𝐵 ) ) )
28 18 27 stoic3 ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω ) → ( ( 𝐴 ·o 𝐵 ) ·o suc 𝑦 ) = ( ( ( 𝐴 ·o 𝐵 ) ·o 𝑦 ) +o ( 𝐴 ·o 𝐵 ) ) )
29 nnmsuc ⊢ ( ( 𝐵 ∈ ω ∧ 𝑦 ∈ ω ) → ( 𝐵 ·o suc 𝑦 ) = ( ( 𝐵 ·o 𝑦 ) +o 𝐵 ) )
30 29 3adant1 ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω ) → ( 𝐵 ·o suc 𝑦 ) = ( ( 𝐵 ·o 𝑦 ) +o 𝐵 ) )
31 30 oveq2d ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω ) → ( 𝐴 ·o ( 𝐵 ·o suc 𝑦 ) ) = ( 𝐴 ·o ( ( 𝐵 ·o 𝑦 ) +o 𝐵 ) ) )
32 nnmcl ⊢ ( ( 𝐵 ∈ ω ∧ 𝑦 ∈ ω ) → ( 𝐵 ·o 𝑦 ) ∈ ω )
33 nndi ⊢ ( ( 𝐴 ∈ ω ∧ ( 𝐵 ·o 𝑦 ) ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ·o ( ( 𝐵 ·o 𝑦 ) +o 𝐵 ) ) = ( ( 𝐴 ·o ( 𝐵 ·o 𝑦 ) ) +o ( 𝐴 ·o 𝐵 ) ) )
34 32 33 syl3an2 ⊢ ( ( 𝐴 ∈ ω ∧ ( 𝐵 ∈ ω ∧ 𝑦 ∈ ω ) ∧ 𝐵 ∈ ω ) → ( 𝐴 ·o ( ( 𝐵 ·o 𝑦 ) +o 𝐵 ) ) = ( ( 𝐴 ·o ( 𝐵 ·o 𝑦 ) ) +o ( 𝐴 ·o 𝐵 ) ) )
35 34 3exp ⊢ ( 𝐴 ∈ ω → ( ( 𝐵 ∈ ω ∧ 𝑦 ∈ ω ) → ( 𝐵 ∈ ω → ( 𝐴 ·o ( ( 𝐵 ·o 𝑦 ) +o 𝐵 ) ) = ( ( 𝐴 ·o ( 𝐵 ·o 𝑦 ) ) +o ( 𝐴 ·o 𝐵 ) ) ) ) )
36 35 expd ⊢ ( 𝐴 ∈ ω → ( 𝐵 ∈ ω → ( 𝑦 ∈ ω → ( 𝐵 ∈ ω → ( 𝐴 ·o ( ( 𝐵 ·o 𝑦 ) +o 𝐵 ) ) = ( ( 𝐴 ·o ( 𝐵 ·o 𝑦 ) ) +o ( 𝐴 ·o 𝐵 ) ) ) ) ) )
37 36 com34 ⊢ ( 𝐴 ∈ ω → ( 𝐵 ∈ ω → ( 𝐵 ∈ ω → ( 𝑦 ∈ ω → ( 𝐴 ·o ( ( 𝐵 ·o 𝑦 ) +o 𝐵 ) ) = ( ( 𝐴 ·o ( 𝐵 ·o 𝑦 ) ) +o ( 𝐴 ·o 𝐵 ) ) ) ) ) )
38 37 pm2.43d ⊢ ( 𝐴 ∈ ω → ( 𝐵 ∈ ω → ( 𝑦 ∈ ω → ( 𝐴 ·o ( ( 𝐵 ·o 𝑦 ) +o 𝐵 ) ) = ( ( 𝐴 ·o ( 𝐵 ·o 𝑦 ) ) +o ( 𝐴 ·o 𝐵 ) ) ) ) )
39 38 3imp ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω ) → ( 𝐴 ·o ( ( 𝐵 ·o 𝑦 ) +o 𝐵 ) ) = ( ( 𝐴 ·o ( 𝐵 ·o 𝑦 ) ) +o ( 𝐴 ·o 𝐵 ) ) )
40 31 39 eqtrd ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω ) → ( 𝐴 ·o ( 𝐵 ·o suc 𝑦 ) ) = ( ( 𝐴 ·o ( 𝐵 ·o 𝑦 ) ) +o ( 𝐴 ·o 𝐵 ) ) )
41 28 40 eqeq12d ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω ) → ( ( ( 𝐴 ·o 𝐵 ) ·o suc 𝑦 ) = ( 𝐴 ·o ( 𝐵 ·o suc 𝑦 ) ) ↔ ( ( ( 𝐴 ·o 𝐵 ) ·o 𝑦 ) +o ( 𝐴 ·o 𝐵 ) ) = ( ( 𝐴 ·o ( 𝐵 ·o 𝑦 ) ) +o ( 𝐴 ·o 𝐵 ) ) ) )
42 26 41 syl5ibr ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω ) → ( ( ( 𝐴 ·o 𝐵 ) ·o 𝑦 ) = ( 𝐴 ·o ( 𝐵 ·o 𝑦 ) ) → ( ( 𝐴 ·o 𝐵 ) ·o suc 𝑦 ) = ( 𝐴 ·o ( 𝐵 ·o suc 𝑦 ) ) ) )
43 42 3exp ⊢ ( 𝐴 ∈ ω → ( 𝐵 ∈ ω → ( 𝑦 ∈ ω → ( ( ( 𝐴 ·o 𝐵 ) ·o 𝑦 ) = ( 𝐴 ·o ( 𝐵 ·o 𝑦 ) ) → ( ( 𝐴 ·o 𝐵 ) ·o suc 𝑦 ) = ( 𝐴 ·o ( 𝐵 ·o suc 𝑦 ) ) ) ) ) )
44 43 com3r ⊢ ( 𝑦 ∈ ω → ( 𝐴 ∈ ω → ( 𝐵 ∈ ω → ( ( ( 𝐴 ·o 𝐵 ) ·o 𝑦 ) = ( 𝐴 ·o ( 𝐵 ·o 𝑦 ) ) → ( ( 𝐴 ·o 𝐵 ) ·o suc 𝑦 ) = ( 𝐴 ·o ( 𝐵 ·o suc 𝑦 ) ) ) ) ) )
45 44 impd ⊢ ( 𝑦 ∈ ω → ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( ( ( 𝐴 ·o 𝐵 ) ·o 𝑦 ) = ( 𝐴 ·o ( 𝐵 ·o 𝑦 ) ) → ( ( 𝐴 ·o 𝐵 ) ·o suc 𝑦 ) = ( 𝐴 ·o ( 𝐵 ·o suc 𝑦 ) ) ) ) )
46 9 13 17 25 45 finds2 ⊢ ( 𝑥 ∈ ω → ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( ( 𝐴 ·o 𝐵 ) ·o 𝑥 ) = ( 𝐴 ·o ( 𝐵 ·o 𝑥 ) ) ) )
47 5 46 vtoclga ⊢ ( 𝐶 ∈ ω → ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( ( 𝐴 ·o 𝐵 ) ·o 𝐶 ) = ( 𝐴 ·o ( 𝐵 ·o 𝐶 ) ) ) )
48 47 expdcom ⊢ ( 𝐴 ∈ ω → ( 𝐵 ∈ ω → ( 𝐶 ∈ ω → ( ( 𝐴 ·o 𝐵 ) ·o 𝐶 ) = ( 𝐴 ·o ( 𝐵 ·o 𝐶 ) ) ) ) )
49 48 3imp ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω ) → ( ( 𝐴 ·o 𝐵 ) ·o 𝐶 ) = ( 𝐴 ·o ( 𝐵 ·o 𝐶 ) ) )