Step | Hyp | Ref
| Expression |
1 | | eqidd 2740 |
. 2
⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑆)) |
2 | | eqidd 2740 |
. 2
⊢ (𝜑 → (+g‘𝑆) = (+g‘𝑆)) |
3 | | psrring.s |
. . 3
⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
4 | | psrring.i |
. . 3
⊢ (𝜑 → 𝐼 ∈ 𝑉) |
5 | | psrring.r |
. . 3
⊢ (𝜑 → 𝑅 ∈ Ring) |
6 | 3, 4, 5 | psrsca 21139 |
. 2
⊢ (𝜑 → 𝑅 = (Scalar‘𝑆)) |
7 | | eqidd 2740 |
. 2
⊢ (𝜑 → (
·𝑠 ‘𝑆) = ( ·𝑠
‘𝑆)) |
8 | | eqidd 2740 |
. 2
⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅)) |
9 | | eqidd 2740 |
. 2
⊢ (𝜑 → (+g‘𝑅) = (+g‘𝑅)) |
10 | | eqidd 2740 |
. 2
⊢ (𝜑 → (.r‘𝑅) = (.r‘𝑅)) |
11 | | eqidd 2740 |
. 2
⊢ (𝜑 → (1r‘𝑅) = (1r‘𝑅)) |
12 | | ringgrp 19769 |
. . . 4
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) |
13 | 5, 12 | syl 17 |
. . 3
⊢ (𝜑 → 𝑅 ∈ Grp) |
14 | 3, 4, 13 | psrgrp 21148 |
. 2
⊢ (𝜑 → 𝑆 ∈ Grp) |
15 | | eqid 2739 |
. . 3
⊢ (
·𝑠 ‘𝑆) = ( ·𝑠
‘𝑆) |
16 | | eqid 2739 |
. . 3
⊢
(Base‘𝑅) =
(Base‘𝑅) |
17 | | eqid 2739 |
. . 3
⊢
(Base‘𝑆) =
(Base‘𝑆) |
18 | 5 | 3ad2ant1 1131 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring) |
19 | | simp2 1135 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑅)) |
20 | | simp3 1136 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆)) |
21 | 3, 15, 16, 17, 18, 19, 20 | psrvscacl 21143 |
. 2
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥( ·𝑠
‘𝑆)𝑦) ∈ (Base‘𝑆)) |
22 | | ovex 7301 |
. . . . . . 7
⊢
(ℕ0 ↑m 𝐼) ∈ V |
23 | 22 | rabex 5259 |
. . . . . 6
⊢ {𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} ∈
V |
24 | 23 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → {𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} ∈
V) |
25 | | simpr1 1192 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑅)) |
26 | | fconst6g 6659 |
. . . . . 6
⊢ (𝑥 ∈ (Base‘𝑅) → ({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}):{𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈
Fin}⟶(Base‘𝑅)) |
27 | 25, 26 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}):{𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈
Fin}⟶(Base‘𝑅)) |
28 | | eqid 2739 |
. . . . . 6
⊢ {𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈
Fin} |
29 | | simpr2 1193 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆)) |
30 | 3, 16, 28, 17, 29 | psrelbas 21129 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦:{𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈
Fin}⟶(Base‘𝑅)) |
31 | | simpr3 1194 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆)) |
32 | 3, 16, 28, 17, 31 | psrelbas 21129 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧:{𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈
Fin}⟶(Base‘𝑅)) |
33 | 5 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Ring) |
34 | | eqid 2739 |
. . . . . . 7
⊢
(+g‘𝑅) = (+g‘𝑅) |
35 | | eqid 2739 |
. . . . . . 7
⊢
(.r‘𝑅) = (.r‘𝑅) |
36 | 16, 34, 35 | ringdi 19786 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → (𝑟(.r‘𝑅)(𝑠(+g‘𝑅)𝑡)) = ((𝑟(.r‘𝑅)𝑠)(+g‘𝑅)(𝑟(.r‘𝑅)𝑡))) |
37 | 33, 36 | sylan 579 |
. . . . 5
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → (𝑟(.r‘𝑅)(𝑠(+g‘𝑅)𝑡)) = ((𝑟(.r‘𝑅)𝑠)(+g‘𝑅)(𝑟(.r‘𝑅)𝑡))) |
38 | 24, 27, 30, 32, 37 | caofdi 7563 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)(𝑦 ∘f
(+g‘𝑅)𝑧)) = ((({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)𝑦) ∘f
(+g‘𝑅)(({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)𝑧))) |
39 | | eqid 2739 |
. . . . . 6
⊢
(+g‘𝑆) = (+g‘𝑆) |
40 | 3, 17, 34, 39, 29, 31 | psradd 21132 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g‘𝑆)𝑧) = (𝑦 ∘f
(+g‘𝑅)𝑧)) |
41 | 40 | oveq2d 7284 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)(𝑦(+g‘𝑆)𝑧)) = (({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)(𝑦 ∘f
(+g‘𝑅)𝑧))) |
42 | 3, 15, 16, 17, 35, 28, 25, 29 | psrvsca 21141 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠
‘𝑆)𝑦) = (({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)𝑦)) |
43 | 3, 15, 16, 17, 35, 28, 25, 31 | psrvsca 21141 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠
‘𝑆)𝑧) = (({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)𝑧)) |
44 | 42, 43 | oveq12d 7286 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥( ·𝑠
‘𝑆)𝑦) ∘f
(+g‘𝑅)(𝑥( ·𝑠
‘𝑆)𝑧)) = ((({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)𝑦) ∘f
(+g‘𝑅)(({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)𝑧))) |
45 | 38, 41, 44 | 3eqtr4d 2789 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)(𝑦(+g‘𝑆)𝑧)) = ((𝑥( ·𝑠
‘𝑆)𝑦) ∘f
(+g‘𝑅)(𝑥( ·𝑠
‘𝑆)𝑧))) |
46 | 13 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Grp) |
47 | 3, 17, 39, 46, 29, 31 | psraddcl 21133 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g‘𝑆)𝑧) ∈ (Base‘𝑆)) |
48 | 3, 15, 16, 17, 35, 28, 25, 47 | psrvsca 21141 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠
‘𝑆)(𝑦(+g‘𝑆)𝑧)) = (({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)(𝑦(+g‘𝑆)𝑧))) |
49 | 21 | 3adant3r3 1182 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠
‘𝑆)𝑦) ∈ (Base‘𝑆)) |
50 | 3, 15, 16, 17, 33, 25, 31 | psrvscacl 21143 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠
‘𝑆)𝑧) ∈ (Base‘𝑆)) |
51 | 3, 17, 34, 39, 49, 50 | psradd 21132 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥( ·𝑠
‘𝑆)𝑦)(+g‘𝑆)(𝑥( ·𝑠
‘𝑆)𝑧)) = ((𝑥( ·𝑠
‘𝑆)𝑦) ∘f
(+g‘𝑅)(𝑥( ·𝑠
‘𝑆)𝑧))) |
52 | 45, 48, 51 | 3eqtr4d 2789 |
. 2
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠
‘𝑆)(𝑦(+g‘𝑆)𝑧)) = ((𝑥( ·𝑠
‘𝑆)𝑦)(+g‘𝑆)(𝑥( ·𝑠
‘𝑆)𝑧))) |
53 | | simpr1 1192 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑅)) |
54 | | simpr3 1194 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆)) |
55 | 3, 15, 16, 17, 35, 28, 53, 54 | psrvsca 21141 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠
‘𝑆)𝑧) = (({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)𝑧)) |
56 | | simpr2 1193 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑅)) |
57 | 3, 15, 16, 17, 35, 28, 56, 54 | psrvsca 21141 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦( ·𝑠
‘𝑆)𝑧) = (({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦}) ∘f
(.r‘𝑅)𝑧)) |
58 | 55, 57 | oveq12d 7286 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥( ·𝑠
‘𝑆)𝑧) ∘f
(+g‘𝑅)(𝑦( ·𝑠
‘𝑆)𝑧)) = ((({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)𝑧) ∘f
(+g‘𝑅)(({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦}) ∘f
(.r‘𝑅)𝑧))) |
59 | 23 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → {𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} ∈
V) |
60 | 3, 16, 28, 17, 54 | psrelbas 21129 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧:{𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈
Fin}⟶(Base‘𝑅)) |
61 | 53, 26 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}):{𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈
Fin}⟶(Base‘𝑅)) |
62 | | fconst6g 6659 |
. . . . . 6
⊢ (𝑦 ∈ (Base‘𝑅) → ({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦}):{𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈
Fin}⟶(Base‘𝑅)) |
63 | 56, 62 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦}):{𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈
Fin}⟶(Base‘𝑅)) |
64 | 5 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Ring) |
65 | 16, 34, 35 | ringdir 19787 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → ((𝑟(+g‘𝑅)𝑠)(.r‘𝑅)𝑡) = ((𝑟(.r‘𝑅)𝑡)(+g‘𝑅)(𝑠(.r‘𝑅)𝑡))) |
66 | 64, 65 | sylan 579 |
. . . . 5
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → ((𝑟(+g‘𝑅)𝑠)(.r‘𝑅)𝑡) = ((𝑟(.r‘𝑅)𝑡)(+g‘𝑅)(𝑠(.r‘𝑅)𝑡))) |
67 | 59, 60, 61, 63, 66 | caofdir 7564 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f
(+g‘𝑅)({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦})) ∘f
(.r‘𝑅)𝑧) = ((({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)𝑧) ∘f
(+g‘𝑅)(({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦}) ∘f
(.r‘𝑅)𝑧))) |
68 | 59, 53, 56 | ofc12 7552 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f
(+g‘𝑅)({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦})) = ({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {(𝑥(+g‘𝑅)𝑦)})) |
69 | 68 | oveq1d 7283 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f
(+g‘𝑅)({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦})) ∘f
(.r‘𝑅)𝑧) = (({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {(𝑥(+g‘𝑅)𝑦)}) ∘f
(.r‘𝑅)𝑧)) |
70 | 58, 67, 69 | 3eqtr2rd 2786 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {(𝑥(+g‘𝑅)𝑦)}) ∘f
(.r‘𝑅)𝑧) = ((𝑥( ·𝑠
‘𝑆)𝑧) ∘f
(+g‘𝑅)(𝑦( ·𝑠
‘𝑆)𝑧))) |
71 | 16, 34 | ringacl 19798 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅)) |
72 | 64, 53, 56, 71 | syl3anc 1369 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g‘𝑅)𝑦) ∈ (Base‘𝑅)) |
73 | 3, 15, 16, 17, 35, 28, 72, 54 | psrvsca 21141 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g‘𝑅)𝑦)( ·𝑠
‘𝑆)𝑧) = (({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {(𝑥(+g‘𝑅)𝑦)}) ∘f
(.r‘𝑅)𝑧)) |
74 | 3, 15, 16, 17, 64, 53, 54 | psrvscacl 21143 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠
‘𝑆)𝑧) ∈ (Base‘𝑆)) |
75 | 3, 15, 16, 17, 64, 56, 54 | psrvscacl 21143 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦( ·𝑠
‘𝑆)𝑧) ∈ (Base‘𝑆)) |
76 | 3, 17, 34, 39, 74, 75 | psradd 21132 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥( ·𝑠
‘𝑆)𝑧)(+g‘𝑆)(𝑦( ·𝑠
‘𝑆)𝑧)) = ((𝑥( ·𝑠
‘𝑆)𝑧) ∘f
(+g‘𝑅)(𝑦( ·𝑠
‘𝑆)𝑧))) |
77 | 70, 73, 76 | 3eqtr4d 2789 |
. 2
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g‘𝑅)𝑦)( ·𝑠
‘𝑆)𝑧) = ((𝑥( ·𝑠
‘𝑆)𝑧)(+g‘𝑆)(𝑦( ·𝑠
‘𝑆)𝑧))) |
78 | 57 | oveq2d 7284 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)(𝑦( ·𝑠
‘𝑆)𝑧)) = (({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)(({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦}) ∘f
(.r‘𝑅)𝑧))) |
79 | 16, 35 | ringass 19784 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → ((𝑟(.r‘𝑅)𝑠)(.r‘𝑅)𝑡) = (𝑟(.r‘𝑅)(𝑠(.r‘𝑅)𝑡))) |
80 | 64, 79 | sylan 579 |
. . . . 5
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → ((𝑟(.r‘𝑅)𝑠)(.r‘𝑅)𝑡) = (𝑟(.r‘𝑅)(𝑠(.r‘𝑅)𝑡))) |
81 | 59, 61, 63, 60, 80 | caofass 7561 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦})) ∘f
(.r‘𝑅)𝑧) = (({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)(({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦}) ∘f
(.r‘𝑅)𝑧))) |
82 | 59, 53, 56 | ofc12 7552 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦})) = ({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {(𝑥(.r‘𝑅)𝑦)})) |
83 | 82 | oveq1d 7283 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑦})) ∘f
(.r‘𝑅)𝑧) = (({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {(𝑥(.r‘𝑅)𝑦)}) ∘f
(.r‘𝑅)𝑧)) |
84 | 78, 81, 83 | 3eqtr2rd 2786 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {(𝑥(.r‘𝑅)𝑦)}) ∘f
(.r‘𝑅)𝑧) = (({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)(𝑦( ·𝑠
‘𝑆)𝑧))) |
85 | 16, 35 | ringcl 19781 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)𝑦) ∈ (Base‘𝑅)) |
86 | 64, 53, 56, 85 | syl3anc 1369 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(.r‘𝑅)𝑦) ∈ (Base‘𝑅)) |
87 | 3, 15, 16, 17, 35, 28, 86, 54 | psrvsca 21141 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(.r‘𝑅)𝑦)( ·𝑠
‘𝑆)𝑧) = (({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {(𝑥(.r‘𝑅)𝑦)}) ∘f
(.r‘𝑅)𝑧)) |
88 | 3, 15, 16, 17, 35, 28, 53, 75 | psrvsca 21141 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠
‘𝑆)(𝑦(
·𝑠 ‘𝑆)𝑧)) = (({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)(𝑦( ·𝑠
‘𝑆)𝑧))) |
89 | 84, 87, 88 | 3eqtr4d 2789 |
. 2
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(.r‘𝑅)𝑦)( ·𝑠
‘𝑆)𝑧) = (𝑥( ·𝑠
‘𝑆)(𝑦(
·𝑠 ‘𝑆)𝑧))) |
90 | 5 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring) |
91 | | eqid 2739 |
. . . . . 6
⊢
(1r‘𝑅) = (1r‘𝑅) |
92 | 16, 91 | ringidcl 19788 |
. . . . 5
⊢ (𝑅 ∈ Ring →
(1r‘𝑅)
∈ (Base‘𝑅)) |
93 | 90, 92 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → (1r‘𝑅) ∈ (Base‘𝑅)) |
94 | | simpr 484 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆)) |
95 | 3, 15, 16, 17, 35, 28, 93, 94 | psrvsca 21141 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → ((1r‘𝑅)(
·𝑠 ‘𝑆)𝑥) = (({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} ×
{(1r‘𝑅)})
∘f (.r‘𝑅)𝑥)) |
96 | 23 | a1i 11 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → {𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} ∈
V) |
97 | 3, 16, 28, 17, 94 | psrelbas 21129 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥:{𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈
Fin}⟶(Base‘𝑅)) |
98 | 16, 35, 91 | ringlidm 19791 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) →
((1r‘𝑅)(.r‘𝑅)𝑟) = 𝑟) |
99 | 90, 98 | sylan 579 |
. . . 4
⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑟 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)𝑟) = 𝑟) |
100 | 96, 97, 93, 99 | caofid0l 7555 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → (({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} ×
{(1r‘𝑅)})
∘f (.r‘𝑅)𝑥) = 𝑥) |
101 | 95, 100 | eqtrd 2779 |
. 2
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → ((1r‘𝑅)(
·𝑠 ‘𝑆)𝑥) = 𝑥) |
102 | 1, 2, 6, 7, 8, 9, 10, 11, 5, 14, 21, 52, 77, 89, 101 | islmodd 20110 |
1
⊢ (𝜑 → 𝑆 ∈ LMod) |