Step | Hyp | Ref
| Expression |
1 | | eqidd 2732 |
. 2
⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑆)) |
2 | | eqidd 2732 |
. 2
⊢ (𝜑 → (+g‘𝑆) = (+g‘𝑆)) |
3 | | psrgrp.s |
. . 3
⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
4 | | eqid 2731 |
. . 3
⊢
(Base‘𝑆) =
(Base‘𝑆) |
5 | | eqid 2731 |
. . 3
⊢
(+g‘𝑆) = (+g‘𝑆) |
6 | | psrgrp.r |
. . . . 5
⊢ (𝜑 → 𝑅 ∈ Grp) |
7 | 6 | grpmgmd 18889 |
. . . 4
⊢ (𝜑 → 𝑅 ∈ Mgm) |
8 | 7 | 3ad2ant1 1132 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ Mgm) |
9 | | simp2 1136 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆)) |
10 | | simp3 1137 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆)) |
11 | 3, 4, 5, 8, 9, 10 | psraddcl 21812 |
. 2
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆)) |
12 | | ovex 7445 |
. . . . . . 7
⊢
(ℕ0 ↑m 𝐼) ∈ V |
13 | 12 | rabex 5332 |
. . . . . 6
⊢ {𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} ∈
V |
14 | 13 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → {𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} ∈
V) |
15 | | eqid 2731 |
. . . . . 6
⊢
(Base‘𝑅) =
(Base‘𝑅) |
16 | | eqid 2731 |
. . . . . 6
⊢ {𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈
Fin} |
17 | | simpr1 1193 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑆)) |
18 | 3, 15, 16, 4, 17 | psrelbas 21808 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥:{𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈
Fin}⟶(Base‘𝑅)) |
19 | | simpr2 1194 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆)) |
20 | 3, 15, 16, 4, 19 | psrelbas 21808 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦:{𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈
Fin}⟶(Base‘𝑅)) |
21 | | simpr3 1195 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆)) |
22 | 3, 15, 16, 4, 21 | psrelbas 21808 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧:{𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈
Fin}⟶(Base‘𝑅)) |
23 | 6 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Grp) |
24 | | eqid 2731 |
. . . . . . 7
⊢
(+g‘𝑅) = (+g‘𝑅) |
25 | 15, 24 | grpass 18870 |
. . . . . 6
⊢ ((𝑅 ∈ Grp ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → ((𝑟(+g‘𝑅)𝑠)(+g‘𝑅)𝑡) = (𝑟(+g‘𝑅)(𝑠(+g‘𝑅)𝑡))) |
26 | 23, 25 | sylan 579 |
. . . . 5
⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → ((𝑟(+g‘𝑅)𝑠)(+g‘𝑅)𝑡) = (𝑟(+g‘𝑅)(𝑠(+g‘𝑅)𝑡))) |
27 | 14, 18, 20, 22, 26 | caofass 7711 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥 ∘f
(+g‘𝑅)𝑦) ∘f
(+g‘𝑅)𝑧) = (𝑥 ∘f
(+g‘𝑅)(𝑦 ∘f
(+g‘𝑅)𝑧))) |
28 | 3, 4, 24, 5, 17, 19 | psradd 21811 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g‘𝑆)𝑦) = (𝑥 ∘f
(+g‘𝑅)𝑦)) |
29 | 28 | oveq1d 7427 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g‘𝑆)𝑦) ∘f
(+g‘𝑅)𝑧) = ((𝑥 ∘f
(+g‘𝑅)𝑦) ∘f
(+g‘𝑅)𝑧)) |
30 | 3, 4, 24, 5, 19, 21 | psradd 21811 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g‘𝑆)𝑧) = (𝑦 ∘f
(+g‘𝑅)𝑧)) |
31 | 30 | oveq2d 7428 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥 ∘f
(+g‘𝑅)(𝑦(+g‘𝑆)𝑧)) = (𝑥 ∘f
(+g‘𝑅)(𝑦 ∘f
(+g‘𝑅)𝑧))) |
32 | 27, 29, 31 | 3eqtr4d 2781 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g‘𝑆)𝑦) ∘f
(+g‘𝑅)𝑧) = (𝑥 ∘f
(+g‘𝑅)(𝑦(+g‘𝑆)𝑧))) |
33 | 11 | 3adant3r3 1183 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆)) |
34 | 3, 4, 24, 5, 33, 21 | psradd 21811 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g‘𝑆)𝑦)(+g‘𝑆)𝑧) = ((𝑥(+g‘𝑆)𝑦) ∘f
(+g‘𝑅)𝑧)) |
35 | 7 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Mgm) |
36 | 3, 4, 5, 35, 19, 21 | psraddcl 21812 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g‘𝑆)𝑧) ∈ (Base‘𝑆)) |
37 | 3, 4, 24, 5, 17, 36 | psradd 21811 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g‘𝑆)(𝑦(+g‘𝑆)𝑧)) = (𝑥 ∘f
(+g‘𝑅)(𝑦(+g‘𝑆)𝑧))) |
38 | 32, 34, 37 | 3eqtr4d 2781 |
. 2
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g‘𝑆)𝑦)(+g‘𝑆)𝑧) = (𝑥(+g‘𝑆)(𝑦(+g‘𝑆)𝑧))) |
39 | | psrgrp.i |
. . 3
⊢ (𝜑 → 𝐼 ∈ 𝑉) |
40 | | eqid 2731 |
. . 3
⊢
(0g‘𝑅) = (0g‘𝑅) |
41 | 3, 39, 6, 16, 40, 4 | psr0cl 21824 |
. 2
⊢ (𝜑 → ({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} ×
{(0g‘𝑅)})
∈ (Base‘𝑆)) |
42 | 39 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝐼 ∈ 𝑉) |
43 | 6 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑅 ∈ Grp) |
44 | | simpr 484 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆)) |
45 | 3, 42, 43, 16, 40, 4, 5, 44 | psr0lid 21825 |
. 2
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → (({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} ×
{(0g‘𝑅)})(+g‘𝑆)𝑥) = 𝑥) |
46 | | eqid 2731 |
. . 3
⊢
(invg‘𝑅) = (invg‘𝑅) |
47 | 3, 42, 43, 16, 46, 4, 44 | psrnegcl 21826 |
. 2
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → ((invg‘𝑅) ∘ 𝑥) ∈ (Base‘𝑆)) |
48 | 3, 42, 43, 16, 46, 4, 44, 40, 5 | psrlinv 21827 |
. 2
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → (((invg‘𝑅) ∘ 𝑥)(+g‘𝑆)𝑥) = ({𝑓 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑓 “ ℕ) ∈ Fin} ×
{(0g‘𝑅)})) |
49 | 1, 2, 11, 38, 41, 45, 47, 48 | isgrpd 18886 |
1
⊢ (𝜑 → 𝑆 ∈ Grp) |