Step | Hyp | Ref
| Expression |
1 | | imasval.u |
. 2
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) |
2 | | df-iimas 12729 |
. . . 4
⊢
“s = (𝑓 ∈ V, 𝑟 ∈ V ↦
⦋(Base‘𝑟) / 𝑣⦌{〈(Base‘ndx), ran
𝑓〉,
〈(+g‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑓‘(𝑝(+g‘𝑟)𝑞))〉}〉,
〈(.r‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑓‘(𝑝(.r‘𝑟)𝑞))〉}〉}) |
3 | 2 | a1i 9 |
. . 3
⊢ (𝜑 → “s
= (𝑓 ∈ V, 𝑟 ∈ V ↦
⦋(Base‘𝑟) / 𝑣⦌{〈(Base‘ndx), ran
𝑓〉,
〈(+g‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑓‘(𝑝(+g‘𝑟)𝑞))〉}〉,
〈(.r‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑓‘(𝑝(.r‘𝑟)𝑞))〉}〉})) |
4 | | basfn 12523 |
. . . . . 6
⊢ Base Fn
V |
5 | | vex 2742 |
. . . . . 6
⊢ 𝑟 ∈ V |
6 | | funfvex 5534 |
. . . . . . 7
⊢ ((Fun
Base ∧ 𝑟 ∈ dom
Base) → (Base‘𝑟)
∈ V) |
7 | 6 | funfni 5318 |
. . . . . 6
⊢ ((Base Fn
V ∧ 𝑟 ∈ V) →
(Base‘𝑟) ∈
V) |
8 | 4, 5, 7 | mp2an 426 |
. . . . 5
⊢
(Base‘𝑟)
∈ V |
9 | 8 | a1i 9 |
. . . 4
⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) → (Base‘𝑟) ∈ V) |
10 | | simplrl 535 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → 𝑓 = 𝐹) |
11 | 10 | rneqd 4858 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ran 𝑓 = ran 𝐹) |
12 | | imasval.f |
. . . . . . . . 9
⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) |
13 | | forn 5443 |
. . . . . . . . 9
⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵) |
14 | 12, 13 | syl 14 |
. . . . . . . 8
⊢ (𝜑 → ran 𝐹 = 𝐵) |
15 | 14 | ad2antrr 488 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ran 𝐹 = 𝐵) |
16 | 11, 15 | eqtrd 2210 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ran 𝑓 = 𝐵) |
17 | 16 | opeq2d 3787 |
. . . . 5
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → 〈(Base‘ndx), ran 𝑓〉 = 〈(Base‘ndx),
𝐵〉) |
18 | | simplrr 536 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → 𝑟 = 𝑅) |
19 | 18 | fveq2d 5521 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → (Base‘𝑟) = (Base‘𝑅)) |
20 | | simpr 110 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → 𝑣 = (Base‘𝑟)) |
21 | | imasval.v |
. . . . . . . . . 10
⊢ (𝜑 → 𝑉 = (Base‘𝑅)) |
22 | 21 | ad2antrr 488 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → 𝑉 = (Base‘𝑅)) |
23 | 19, 20, 22 | 3eqtr4d 2220 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → 𝑣 = 𝑉) |
24 | 10 | fveq1d 5519 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → (𝑓‘𝑝) = (𝐹‘𝑝)) |
25 | 10 | fveq1d 5519 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → (𝑓‘𝑞) = (𝐹‘𝑞)) |
26 | 24, 25 | opeq12d 3788 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → 〈(𝑓‘𝑝), (𝑓‘𝑞)〉 = 〈(𝐹‘𝑝), (𝐹‘𝑞)〉) |
27 | 18 | fveq2d 5521 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → (+g‘𝑟) = (+g‘𝑅)) |
28 | | imasval.p |
. . . . . . . . . . . . . 14
⊢ + =
(+g‘𝑅) |
29 | 27, 28 | eqtr4di 2228 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → (+g‘𝑟) = + ) |
30 | 29 | oveqd 5895 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → (𝑝(+g‘𝑟)𝑞) = (𝑝 + 𝑞)) |
31 | 10, 30 | fveq12d 5524 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → (𝑓‘(𝑝(+g‘𝑟)𝑞)) = (𝐹‘(𝑝 + 𝑞))) |
32 | 26, 31 | opeq12d 3788 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → 〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑓‘(𝑝(+g‘𝑟)𝑞))〉 = 〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 + 𝑞))〉) |
33 | 32 | sneqd 3607 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → {〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑓‘(𝑝(+g‘𝑟)𝑞))〉} = {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 + 𝑞))〉}) |
34 | 23, 33 | iuneq12d 3912 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ∪
𝑞 ∈ 𝑣 {〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑓‘(𝑝(+g‘𝑟)𝑞))〉} = ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 + 𝑞))〉}) |
35 | 23, 34 | iuneq12d 3912 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ∪
𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑓‘(𝑝(+g‘𝑟)𝑞))〉} = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 + 𝑞))〉}) |
36 | | imasval.a |
. . . . . . . 8
⊢ (𝜑 → ✚ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 + 𝑞))〉}) |
37 | 36 | ad2antrr 488 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ✚ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 + 𝑞))〉}) |
38 | 35, 37 | eqtr4d 2213 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ∪
𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑓‘(𝑝(+g‘𝑟)𝑞))〉} = ✚ ) |
39 | 38 | opeq2d 3787 |
. . . . 5
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → 〈(+g‘ndx),
∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑓‘(𝑝(+g‘𝑟)𝑞))〉}〉 =
〈(+g‘ndx), ✚
〉) |
40 | 18 | fveq2d 5521 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → (.r‘𝑟) = (.r‘𝑅)) |
41 | | imasval.m |
. . . . . . . . . . . . . 14
⊢ × =
(.r‘𝑅) |
42 | 40, 41 | eqtr4di 2228 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → (.r‘𝑟) = × ) |
43 | 42 | oveqd 5895 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → (𝑝(.r‘𝑟)𝑞) = (𝑝 × 𝑞)) |
44 | 10, 43 | fveq12d 5524 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → (𝑓‘(𝑝(.r‘𝑟)𝑞)) = (𝐹‘(𝑝 × 𝑞))) |
45 | 26, 44 | opeq12d 3788 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → 〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑓‘(𝑝(.r‘𝑟)𝑞))〉 = 〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 × 𝑞))〉) |
46 | 45 | sneqd 3607 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → {〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑓‘(𝑝(.r‘𝑟)𝑞))〉} = {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 × 𝑞))〉}) |
47 | 23, 46 | iuneq12d 3912 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ∪
𝑞 ∈ 𝑣 {〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑓‘(𝑝(.r‘𝑟)𝑞))〉} = ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 × 𝑞))〉}) |
48 | 23, 47 | iuneq12d 3912 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ∪
𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑓‘(𝑝(.r‘𝑟)𝑞))〉} = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 × 𝑞))〉}) |
49 | | imasval.t |
. . . . . . . 8
⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 × 𝑞))〉}) |
50 | 49 | ad2antrr 488 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 × 𝑞))〉}) |
51 | 48, 50 | eqtr4d 2213 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → ∪
𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑓‘(𝑝(.r‘𝑟)𝑞))〉} = ∙ ) |
52 | 51 | opeq2d 3787 |
. . . . 5
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → 〈(.r‘ndx),
∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑓‘(𝑝(.r‘𝑟)𝑞))〉}〉 =
〈(.r‘ndx), ∙
〉) |
53 | 17, 39, 52 | tpeq123d 3686 |
. . . 4
⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) ∧ 𝑣 = (Base‘𝑟)) → {〈(Base‘ndx), ran 𝑓〉,
〈(+g‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑓‘(𝑝(+g‘𝑟)𝑞))〉}〉,
〈(.r‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑓‘(𝑝(.r‘𝑟)𝑞))〉}〉} = {〈(Base‘ndx),
𝐵〉,
〈(+g‘ndx), ✚ 〉,
〈(.r‘ndx), ∙
〉}) |
54 | 9, 53 | csbied 3105 |
. . 3
⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑟 = 𝑅)) → ⦋(Base‘𝑟) / 𝑣⦌{〈(Base‘ndx), ran
𝑓〉,
〈(+g‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑓‘(𝑝(+g‘𝑟)𝑞))〉}〉,
〈(.r‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑓‘(𝑝(.r‘𝑟)𝑞))〉}〉} = {〈(Base‘ndx),
𝐵〉,
〈(+g‘ndx), ✚ 〉,
〈(.r‘ndx), ∙
〉}) |
55 | | fof 5440 |
. . . . 5
⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵) |
56 | 12, 55 | syl 14 |
. . . 4
⊢ (𝜑 → 𝐹:𝑉⟶𝐵) |
57 | | imasval.r |
. . . . . . 7
⊢ (𝜑 → 𝑅 ∈ 𝑍) |
58 | 57 | elexd 2752 |
. . . . . 6
⊢ (𝜑 → 𝑅 ∈ V) |
59 | | funfvex 5534 |
. . . . . . 7
⊢ ((Fun
Base ∧ 𝑅 ∈ dom
Base) → (Base‘𝑅)
∈ V) |
60 | 59 | funfni 5318 |
. . . . . 6
⊢ ((Base Fn
V ∧ 𝑅 ∈ V) →
(Base‘𝑅) ∈
V) |
61 | 4, 58, 60 | sylancr 414 |
. . . . 5
⊢ (𝜑 → (Base‘𝑅) ∈ V) |
62 | 21, 61 | eqeltrd 2254 |
. . . 4
⊢ (𝜑 → 𝑉 ∈ V) |
63 | 56, 62 | fexd 5749 |
. . 3
⊢ (𝜑 → 𝐹 ∈ V) |
64 | | basendxnn 12521 |
. . . . 5
⊢
(Base‘ndx) ∈ ℕ |
65 | | focdmex 6119 |
. . . . . 6
⊢ (𝑉 ∈ V → (𝐹:𝑉–onto→𝐵 → 𝐵 ∈ V)) |
66 | 62, 12, 65 | sylc 62 |
. . . . 5
⊢ (𝜑 → 𝐵 ∈ V) |
67 | | opexg 4230 |
. . . . 5
⊢
(((Base‘ndx) ∈ ℕ ∧ 𝐵 ∈ V) → 〈(Base‘ndx),
𝐵〉 ∈
V) |
68 | 64, 66, 67 | sylancr 414 |
. . . 4
⊢ (𝜑 → 〈(Base‘ndx),
𝐵〉 ∈
V) |
69 | | plusgndxnn 12573 |
. . . . 5
⊢
(+g‘ndx) ∈ ℕ |
70 | 63 | ad2antrr 488 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → 𝐹 ∈ V) |
71 | | vex 2742 |
. . . . . . . . . . . . . . 15
⊢ 𝑝 ∈ V |
72 | 71 | a1i 9 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → 𝑝 ∈ V) |
73 | | fvexg 5536 |
. . . . . . . . . . . . . 14
⊢ ((𝐹 ∈ V ∧ 𝑝 ∈ V) → (𝐹‘𝑝) ∈ V) |
74 | 70, 72, 73 | syl2anc 411 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → (𝐹‘𝑝) ∈ V) |
75 | | vex 2742 |
. . . . . . . . . . . . . . 15
⊢ 𝑞 ∈ V |
76 | 75 | a1i 9 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → 𝑞 ∈ V) |
77 | | fvexg 5536 |
. . . . . . . . . . . . . 14
⊢ ((𝐹 ∈ V ∧ 𝑞 ∈ V) → (𝐹‘𝑞) ∈ V) |
78 | 70, 76, 77 | syl2anc 411 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → (𝐹‘𝑞) ∈ V) |
79 | | opexg 4230 |
. . . . . . . . . . . . 13
⊢ (((𝐹‘𝑝) ∈ V ∧ (𝐹‘𝑞) ∈ V) → 〈(𝐹‘𝑝), (𝐹‘𝑞)〉 ∈ V) |
80 | 74, 78, 79 | syl2anc 411 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → 〈(𝐹‘𝑝), (𝐹‘𝑞)〉 ∈ V) |
81 | | plusgslid 12574 |
. . . . . . . . . . . . . . . . . 18
⊢
(+g = Slot (+g‘ndx) ∧
(+g‘ndx) ∈ ℕ) |
82 | 81 | slotex 12492 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑅 ∈ 𝑍 → (+g‘𝑅) ∈ V) |
83 | 57, 82 | syl 14 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (+g‘𝑅) ∈ V) |
84 | 28, 83 | eqeltrid 2264 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → + ∈ V) |
85 | 84 | ad2antrr 488 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → + ∈ V) |
86 | | ovexg 5912 |
. . . . . . . . . . . . . 14
⊢ ((𝑝 ∈ V ∧ + ∈ V
∧ 𝑞 ∈ V) →
(𝑝 + 𝑞) ∈ V) |
87 | 72, 85, 76, 86 | syl3anc 1238 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → (𝑝 + 𝑞) ∈ V) |
88 | | fvexg 5536 |
. . . . . . . . . . . . 13
⊢ ((𝐹 ∈ V ∧ (𝑝 + 𝑞) ∈ V) → (𝐹‘(𝑝 + 𝑞)) ∈ V) |
89 | 70, 87, 88 | syl2anc 411 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → (𝐹‘(𝑝 + 𝑞)) ∈ V) |
90 | | opexg 4230 |
. . . . . . . . . . . 12
⊢
((〈(𝐹‘𝑝), (𝐹‘𝑞)〉 ∈ V ∧ (𝐹‘(𝑝 + 𝑞)) ∈ V) → 〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 + 𝑞))〉 ∈ V) |
91 | 80, 89, 90 | syl2anc 411 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → 〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 + 𝑞))〉 ∈ V) |
92 | | snexg 4186 |
. . . . . . . . . . 11
⊢
(〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 + 𝑞))〉 ∈ V → {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 + 𝑞))〉} ∈ V) |
93 | 91, 92 | syl 14 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 + 𝑞))〉} ∈ V) |
94 | 93 | ralrimiva 2550 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑝 ∈ 𝑉) → ∀𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 + 𝑞))〉} ∈ V) |
95 | | iunexg 6123 |
. . . . . . . . 9
⊢ ((𝑉 ∈ V ∧ ∀𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 + 𝑞))〉} ∈ V) → ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 + 𝑞))〉} ∈ V) |
96 | 62, 94, 95 | syl2an2r 595 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑝 ∈ 𝑉) → ∪
𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 + 𝑞))〉} ∈ V) |
97 | 96 | ralrimiva 2550 |
. . . . . . 7
⊢ (𝜑 → ∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 + 𝑞))〉} ∈ V) |
98 | | iunexg 6123 |
. . . . . . 7
⊢ ((𝑉 ∈ V ∧ ∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 + 𝑞))〉} ∈ V) → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 + 𝑞))〉} ∈ V) |
99 | 62, 97, 98 | syl2anc 411 |
. . . . . 6
⊢ (𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 + 𝑞))〉} ∈ V) |
100 | 36, 99 | eqeltrd 2254 |
. . . . 5
⊢ (𝜑 → ✚ ∈
V) |
101 | | opexg 4230 |
. . . . 5
⊢
(((+g‘ndx) ∈ ℕ ∧ ✚ ∈ V) →
〈(+g‘ndx), ✚ 〉 ∈
V) |
102 | 69, 100, 101 | sylancr 414 |
. . . 4
⊢ (𝜑 →
〈(+g‘ndx), ✚ 〉 ∈
V) |
103 | | mulrslid 12593 |
. . . . . 6
⊢
(.r = Slot (.r‘ndx) ∧
(.r‘ndx) ∈ ℕ) |
104 | 103 | simpri 113 |
. . . . 5
⊢
(.r‘ndx) ∈ ℕ |
105 | 103 | slotex 12492 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑅 ∈ 𝑍 → (.r‘𝑅) ∈ V) |
106 | 57, 105 | syl 14 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (.r‘𝑅) ∈ V) |
107 | 41, 106 | eqeltrid 2264 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → × ∈
V) |
108 | 107 | ad2antrr 488 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → × ∈
V) |
109 | | ovexg 5912 |
. . . . . . . . . . . . . 14
⊢ ((𝑝 ∈ V ∧ × ∈
V ∧ 𝑞 ∈ V) →
(𝑝 × 𝑞) ∈ V) |
110 | 72, 108, 76, 109 | syl3anc 1238 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → (𝑝 × 𝑞) ∈ V) |
111 | | fvexg 5536 |
. . . . . . . . . . . . 13
⊢ ((𝐹 ∈ V ∧ (𝑝 × 𝑞) ∈ V) → (𝐹‘(𝑝 × 𝑞)) ∈ V) |
112 | 70, 110, 111 | syl2anc 411 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → (𝐹‘(𝑝 × 𝑞)) ∈ V) |
113 | | opexg 4230 |
. . . . . . . . . . . 12
⊢
((〈(𝐹‘𝑝), (𝐹‘𝑞)〉 ∈ V ∧ (𝐹‘(𝑝 × 𝑞)) ∈ V) → 〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 × 𝑞))〉 ∈ V) |
114 | 80, 112, 113 | syl2anc 411 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → 〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 × 𝑞))〉 ∈ V) |
115 | | snexg 4186 |
. . . . . . . . . . 11
⊢
(〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 × 𝑞))〉 ∈ V → {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 × 𝑞))〉} ∈ V) |
116 | 114, 115 | syl 14 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 × 𝑞))〉} ∈ V) |
117 | 116 | ralrimiva 2550 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑝 ∈ 𝑉) → ∀𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 × 𝑞))〉} ∈ V) |
118 | | iunexg 6123 |
. . . . . . . . 9
⊢ ((𝑉 ∈ V ∧ ∀𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 × 𝑞))〉} ∈ V) → ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 × 𝑞))〉} ∈ V) |
119 | 62, 117, 118 | syl2an2r 595 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑝 ∈ 𝑉) → ∪
𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 × 𝑞))〉} ∈ V) |
120 | 119 | ralrimiva 2550 |
. . . . . . 7
⊢ (𝜑 → ∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 × 𝑞))〉} ∈ V) |
121 | | iunexg 6123 |
. . . . . . 7
⊢ ((𝑉 ∈ V ∧ ∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 × 𝑞))〉} ∈ V) → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 × 𝑞))〉} ∈ V) |
122 | 62, 120, 121 | syl2anc 411 |
. . . . . 6
⊢ (𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 × 𝑞))〉} ∈ V) |
123 | 49, 122 | eqeltrd 2254 |
. . . . 5
⊢ (𝜑 → ∙ ∈
V) |
124 | | opexg 4230 |
. . . . 5
⊢
(((.r‘ndx) ∈ ℕ ∧ ∙ ∈ V) →
〈(.r‘ndx), ∙ 〉 ∈
V) |
125 | 104, 123,
124 | sylancr 414 |
. . . 4
⊢ (𝜑 →
〈(.r‘ndx), ∙ 〉 ∈
V) |
126 | | tpexg 4446 |
. . . 4
⊢
((〈(Base‘ndx), 𝐵〉 ∈ V ∧
〈(+g‘ndx), ✚ 〉 ∈ V
∧ 〈(.r‘ndx), ∙ 〉 ∈ V)
→ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx),
✚
〉, 〈(.r‘ndx), ∙ 〉} ∈
V) |
127 | 68, 102, 125, 126 | syl3anc 1238 |
. . 3
⊢ (𝜑 → {〈(Base‘ndx),
𝐵〉,
〈(+g‘ndx), ✚ 〉,
〈(.r‘ndx), ∙ 〉} ∈
V) |
128 | 3, 54, 63, 58, 127 | ovmpod 6005 |
. 2
⊢ (𝜑 → (𝐹 “s 𝑅) = {〈(Base‘ndx),
𝐵〉,
〈(+g‘ndx), ✚ 〉,
〈(.r‘ndx), ∙
〉}) |
129 | 1, 128 | eqtrd 2210 |
1
⊢ (𝜑 → 𝑈 = {〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), ✚ 〉,
〈(.r‘ndx), ∙
〉}) |