Step | Hyp | Ref
| Expression |
1 | | oveq2 5859 |
. . . . . 6
⊢ (𝑛 = ∅ → (𝐴 +o 𝑛) = (𝐴 +o ∅)) |
2 | 1 | fveq2d 5498 |
. . . . 5
⊢ (𝑛 = ∅ → (𝐺‘(𝐴 +o 𝑛)) = (𝐺‘(𝐴 +o ∅))) |
3 | | fveq2 5494 |
. . . . . 6
⊢ (𝑛 = ∅ → (𝐺‘𝑛) = (𝐺‘∅)) |
4 | 3 | oveq2d 5867 |
. . . . 5
⊢ (𝑛 = ∅ → ((𝐺‘𝐴) + (𝐺‘𝑛)) = ((𝐺‘𝐴) + (𝐺‘∅))) |
5 | 2, 4 | eqeq12d 2185 |
. . . 4
⊢ (𝑛 = ∅ → ((𝐺‘(𝐴 +o 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛)) ↔ (𝐺‘(𝐴 +o ∅)) = ((𝐺‘𝐴) + (𝐺‘∅)))) |
6 | 5 | imbi2d 229 |
. . 3
⊢ (𝑛 = ∅ → ((𝐴 ∈ ω → (𝐺‘(𝐴 +o 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛))) ↔ (𝐴 ∈ ω → (𝐺‘(𝐴 +o ∅)) = ((𝐺‘𝐴) + (𝐺‘∅))))) |
7 | | oveq2 5859 |
. . . . . 6
⊢ (𝑛 = 𝑧 → (𝐴 +o 𝑛) = (𝐴 +o 𝑧)) |
8 | 7 | fveq2d 5498 |
. . . . 5
⊢ (𝑛 = 𝑧 → (𝐺‘(𝐴 +o 𝑛)) = (𝐺‘(𝐴 +o 𝑧))) |
9 | | fveq2 5494 |
. . . . . 6
⊢ (𝑛 = 𝑧 → (𝐺‘𝑛) = (𝐺‘𝑧)) |
10 | 9 | oveq2d 5867 |
. . . . 5
⊢ (𝑛 = 𝑧 → ((𝐺‘𝐴) + (𝐺‘𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) |
11 | 8, 10 | eqeq12d 2185 |
. . . 4
⊢ (𝑛 = 𝑧 → ((𝐺‘(𝐴 +o 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛)) ↔ (𝐺‘(𝐴 +o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧)))) |
12 | 11 | imbi2d 229 |
. . 3
⊢ (𝑛 = 𝑧 → ((𝐴 ∈ ω → (𝐺‘(𝐴 +o 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛))) ↔ (𝐴 ∈ ω → (𝐺‘(𝐴 +o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))))) |
13 | | oveq2 5859 |
. . . . . 6
⊢ (𝑛 = suc 𝑧 → (𝐴 +o 𝑛) = (𝐴 +o suc 𝑧)) |
14 | 13 | fveq2d 5498 |
. . . . 5
⊢ (𝑛 = suc 𝑧 → (𝐺‘(𝐴 +o 𝑛)) = (𝐺‘(𝐴 +o suc 𝑧))) |
15 | | fveq2 5494 |
. . . . . 6
⊢ (𝑛 = suc 𝑧 → (𝐺‘𝑛) = (𝐺‘suc 𝑧)) |
16 | 15 | oveq2d 5867 |
. . . . 5
⊢ (𝑛 = suc 𝑧 → ((𝐺‘𝐴) + (𝐺‘𝑛)) = ((𝐺‘𝐴) + (𝐺‘suc 𝑧))) |
17 | 14, 16 | eqeq12d 2185 |
. . . 4
⊢ (𝑛 = suc 𝑧 → ((𝐺‘(𝐴 +o 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛)) ↔ (𝐺‘(𝐴 +o suc 𝑧)) = ((𝐺‘𝐴) + (𝐺‘suc 𝑧)))) |
18 | 17 | imbi2d 229 |
. . 3
⊢ (𝑛 = suc 𝑧 → ((𝐴 ∈ ω → (𝐺‘(𝐴 +o 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛))) ↔ (𝐴 ∈ ω → (𝐺‘(𝐴 +o suc 𝑧)) = ((𝐺‘𝐴) + (𝐺‘suc 𝑧))))) |
19 | | oveq2 5859 |
. . . . . 6
⊢ (𝑛 = 𝐵 → (𝐴 +o 𝑛) = (𝐴 +o 𝐵)) |
20 | 19 | fveq2d 5498 |
. . . . 5
⊢ (𝑛 = 𝐵 → (𝐺‘(𝐴 +o 𝑛)) = (𝐺‘(𝐴 +o 𝐵))) |
21 | | fveq2 5494 |
. . . . . 6
⊢ (𝑛 = 𝐵 → (𝐺‘𝑛) = (𝐺‘𝐵)) |
22 | 21 | oveq2d 5867 |
. . . . 5
⊢ (𝑛 = 𝐵 → ((𝐺‘𝐴) + (𝐺‘𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝐵))) |
23 | 20, 22 | eqeq12d 2185 |
. . . 4
⊢ (𝑛 = 𝐵 → ((𝐺‘(𝐴 +o 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛)) ↔ (𝐺‘(𝐴 +o 𝐵)) = ((𝐺‘𝐴) + (𝐺‘𝐵)))) |
24 | 23 | imbi2d 229 |
. . 3
⊢ (𝑛 = 𝐵 → ((𝐴 ∈ ω → (𝐺‘(𝐴 +o 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛))) ↔ (𝐴 ∈ ω → (𝐺‘(𝐴 +o 𝐵)) = ((𝐺‘𝐴) + (𝐺‘𝐵))))) |
25 | | omgadd.1 |
. . . . . . . . 9
⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) |
26 | 25 | frechashgf1o 10373 |
. . . . . . . 8
⊢ 𝐺:ω–1-1-onto→ℕ0 |
27 | | f1of 5440 |
. . . . . . . 8
⊢ (𝐺:ω–1-1-onto→ℕ0 → 𝐺:ω⟶ℕ0) |
28 | 26, 27 | ax-mp 5 |
. . . . . . 7
⊢ 𝐺:ω⟶ℕ0 |
29 | 28 | ffvelrni 5628 |
. . . . . 6
⊢ (𝐴 ∈ ω → (𝐺‘𝐴) ∈
ℕ0) |
30 | 29 | nn0cnd 9179 |
. . . . 5
⊢ (𝐴 ∈ ω → (𝐺‘𝐴) ∈ ℂ) |
31 | 30 | addid1d 8057 |
. . . 4
⊢ (𝐴 ∈ ω → ((𝐺‘𝐴) + 0) = (𝐺‘𝐴)) |
32 | | 0zd 9213 |
. . . . . 6
⊢ (𝐴 ∈ ω → 0 ∈
ℤ) |
33 | 32, 25 | frec2uz0d 10344 |
. . . . 5
⊢ (𝐴 ∈ ω → (𝐺‘∅) =
0) |
34 | 33 | oveq2d 5867 |
. . . 4
⊢ (𝐴 ∈ ω → ((𝐺‘𝐴) + (𝐺‘∅)) = ((𝐺‘𝐴) + 0)) |
35 | | nna0 6451 |
. . . . 5
⊢ (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴) |
36 | 35 | fveq2d 5498 |
. . . 4
⊢ (𝐴 ∈ ω → (𝐺‘(𝐴 +o ∅)) = (𝐺‘𝐴)) |
37 | 31, 34, 36 | 3eqtr4rd 2214 |
. . 3
⊢ (𝐴 ∈ ω → (𝐺‘(𝐴 +o ∅)) = ((𝐺‘𝐴) + (𝐺‘∅))) |
38 | | nnasuc 6453 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐴 +o suc 𝑧) = suc (𝐴 +o 𝑧)) |
39 | 38 | fveq2d 5498 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐺‘(𝐴 +o suc 𝑧)) = (𝐺‘suc (𝐴 +o 𝑧))) |
40 | | 0zd 9213 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → 0 ∈
ℤ) |
41 | | nnacl 6457 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐴 +o 𝑧) ∈
ω) |
42 | 40, 25, 41 | frec2uzsucd 10346 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐺‘suc (𝐴 +o 𝑧)) = ((𝐺‘(𝐴 +o 𝑧)) + 1)) |
43 | 39, 42 | eqtrd 2203 |
. . . . . . . 8
⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐺‘(𝐴 +o suc 𝑧)) = ((𝐺‘(𝐴 +o 𝑧)) + 1)) |
44 | 43 | 3adant3 1012 |
. . . . . . 7
⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) → (𝐺‘(𝐴 +o suc 𝑧)) = ((𝐺‘(𝐴 +o 𝑧)) + 1)) |
45 | 30 | 3ad2ant1 1013 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) → (𝐺‘𝐴) ∈ ℂ) |
46 | 28 | ffvelrni 5628 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈
ℕ0) |
47 | 46 | nn0cnd 9179 |
. . . . . . . . . 10
⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ ℂ) |
48 | 47 | 3ad2ant2 1014 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) → (𝐺‘𝑧) ∈ ℂ) |
49 | | 1cnd 7925 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) → 1 ∈ ℂ) |
50 | 45, 48, 49 | addassd 7931 |
. . . . . . . 8
⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) → (((𝐺‘𝐴) + (𝐺‘𝑧)) + 1) = ((𝐺‘𝐴) + ((𝐺‘𝑧) + 1))) |
51 | | oveq1 5858 |
. . . . . . . . 9
⊢ ((𝐺‘(𝐴 +o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧)) → ((𝐺‘(𝐴 +o 𝑧)) + 1) = (((𝐺‘𝐴) + (𝐺‘𝑧)) + 1)) |
52 | 51 | 3ad2ant3 1015 |
. . . . . . . 8
⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) → ((𝐺‘(𝐴 +o 𝑧)) + 1) = (((𝐺‘𝐴) + (𝐺‘𝑧)) + 1)) |
53 | | 0zd 9213 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ ω → 0 ∈
ℤ) |
54 | | id 19 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ ω → 𝑧 ∈
ω) |
55 | 53, 25, 54 | frec2uzsucd 10346 |
. . . . . . . . . 10
⊢ (𝑧 ∈ ω → (𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1)) |
56 | 55 | oveq2d 5867 |
. . . . . . . . 9
⊢ (𝑧 ∈ ω → ((𝐺‘𝐴) + (𝐺‘suc 𝑧)) = ((𝐺‘𝐴) + ((𝐺‘𝑧) + 1))) |
57 | 56 | 3ad2ant2 1014 |
. . . . . . . 8
⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) → ((𝐺‘𝐴) + (𝐺‘suc 𝑧)) = ((𝐺‘𝐴) + ((𝐺‘𝑧) + 1))) |
58 | 50, 52, 57 | 3eqtr4d 2213 |
. . . . . . 7
⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) → ((𝐺‘(𝐴 +o 𝑧)) + 1) = ((𝐺‘𝐴) + (𝐺‘suc 𝑧))) |
59 | 44, 58 | eqtrd 2203 |
. . . . . 6
⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) → (𝐺‘(𝐴 +o suc 𝑧)) = ((𝐺‘𝐴) + (𝐺‘suc 𝑧))) |
60 | 59 | 3expia 1200 |
. . . . 5
⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → ((𝐺‘(𝐴 +o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧)) → (𝐺‘(𝐴 +o suc 𝑧)) = ((𝐺‘𝐴) + (𝐺‘suc 𝑧)))) |
61 | 60 | expcom 115 |
. . . 4
⊢ (𝑧 ∈ ω → (𝐴 ∈ ω → ((𝐺‘(𝐴 +o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧)) → (𝐺‘(𝐴 +o suc 𝑧)) = ((𝐺‘𝐴) + (𝐺‘suc 𝑧))))) |
62 | 61 | a2d 26 |
. . 3
⊢ (𝑧 ∈ ω → ((𝐴 ∈ ω → (𝐺‘(𝐴 +o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) → (𝐴 ∈ ω → (𝐺‘(𝐴 +o suc 𝑧)) = ((𝐺‘𝐴) + (𝐺‘suc 𝑧))))) |
63 | 6, 12, 18, 24, 37, 62 | finds 4582 |
. 2
⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐺‘(𝐴 +o 𝐵)) = ((𝐺‘𝐴) + (𝐺‘𝐵)))) |
64 | 63 | impcom 124 |
1
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐺‘(𝐴 +o 𝐵)) = ((𝐺‘𝐴) + (𝐺‘𝐵))) |