Step | Hyp | Ref
| Expression |
1 | | eqidd 2738 |
. 2
⊢ (𝜑 → (Base‘𝑌) = (Base‘𝑌)) |
2 | | eqidd 2738 |
. 2
⊢ (𝜑 → (+g‘𝑌) = (+g‘𝑌)) |
3 | | prdsmndd.y |
. . . 4
⊢ 𝑌 = (𝑆Xs𝑅) |
4 | | eqid 2737 |
. . . 4
⊢
(Base‘𝑌) =
(Base‘𝑌) |
5 | | eqid 2737 |
. . . 4
⊢
(+g‘𝑌) = (+g‘𝑌) |
6 | | prdsmndd.s |
. . . . . 6
⊢ (𝜑 → 𝑆 ∈ 𝑉) |
7 | 6 | elexd 3428 |
. . . . 5
⊢ (𝜑 → 𝑆 ∈ V) |
8 | 7 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑆 ∈ V) |
9 | | prdsmndd.i |
. . . . . 6
⊢ (𝜑 → 𝐼 ∈ 𝑊) |
10 | 9 | elexd 3428 |
. . . . 5
⊢ (𝜑 → 𝐼 ∈ V) |
11 | 10 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝐼 ∈ V) |
12 | | prdsmndd.r |
. . . . 5
⊢ (𝜑 → 𝑅:𝐼⟶Mnd) |
13 | 12 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Mnd) |
14 | | simprl 771 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑌)) |
15 | | simprr 773 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌)) |
16 | 3, 4, 5, 8, 11, 13, 14, 15 | prdsplusgcl 18204 |
. . 3
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → (𝑎(+g‘𝑌)𝑏) ∈ (Base‘𝑌)) |
17 | 16 | 3impb 1117 |
. 2
⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → (𝑎(+g‘𝑌)𝑏) ∈ (Base‘𝑌)) |
18 | 12 | ffvelrnda 6904 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ Mnd) |
19 | 18 | adantlr 715 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ Mnd) |
20 | 7 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑆 ∈ V) |
21 | 10 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝐼 ∈ V) |
22 | 12 | ffnd 6546 |
. . . . . . . 8
⊢ (𝜑 → 𝑅 Fn 𝐼) |
23 | 22 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑅 Fn 𝐼) |
24 | | simplr1 1217 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑎 ∈ (Base‘𝑌)) |
25 | | simpr 488 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑦 ∈ 𝐼) |
26 | 3, 4, 20, 21, 23, 24, 25 | prdsbasprj 16977 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑎‘𝑦) ∈ (Base‘(𝑅‘𝑦))) |
27 | | simplr2 1218 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑏 ∈ (Base‘𝑌)) |
28 | 3, 4, 20, 21, 23, 27, 25 | prdsbasprj 16977 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑏‘𝑦) ∈ (Base‘(𝑅‘𝑦))) |
29 | | simplr3 1219 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑐 ∈ (Base‘𝑌)) |
30 | 3, 4, 20, 21, 23, 29, 25 | prdsbasprj 16977 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑐‘𝑦) ∈ (Base‘(𝑅‘𝑦))) |
31 | | eqid 2737 |
. . . . . . 7
⊢
(Base‘(𝑅‘𝑦)) = (Base‘(𝑅‘𝑦)) |
32 | | eqid 2737 |
. . . . . . 7
⊢
(+g‘(𝑅‘𝑦)) = (+g‘(𝑅‘𝑦)) |
33 | 31, 32 | mndass 18182 |
. . . . . 6
⊢ (((𝑅‘𝑦) ∈ Mnd ∧ ((𝑎‘𝑦) ∈ (Base‘(𝑅‘𝑦)) ∧ (𝑏‘𝑦) ∈ (Base‘(𝑅‘𝑦)) ∧ (𝑐‘𝑦) ∈ (Base‘(𝑅‘𝑦)))) → (((𝑎‘𝑦)(+g‘(𝑅‘𝑦))(𝑏‘𝑦))(+g‘(𝑅‘𝑦))(𝑐‘𝑦)) = ((𝑎‘𝑦)(+g‘(𝑅‘𝑦))((𝑏‘𝑦)(+g‘(𝑅‘𝑦))(𝑐‘𝑦)))) |
34 | 19, 26, 28, 30, 33 | syl13anc 1374 |
. . . . 5
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (((𝑎‘𝑦)(+g‘(𝑅‘𝑦))(𝑏‘𝑦))(+g‘(𝑅‘𝑦))(𝑐‘𝑦)) = ((𝑎‘𝑦)(+g‘(𝑅‘𝑦))((𝑏‘𝑦)(+g‘(𝑅‘𝑦))(𝑐‘𝑦)))) |
35 | 3, 4, 20, 21, 23, 24, 27, 5, 25 | prdsplusgfval 16979 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑎(+g‘𝑌)𝑏)‘𝑦) = ((𝑎‘𝑦)(+g‘(𝑅‘𝑦))(𝑏‘𝑦))) |
36 | 35 | oveq1d 7228 |
. . . . 5
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (((𝑎(+g‘𝑌)𝑏)‘𝑦)(+g‘(𝑅‘𝑦))(𝑐‘𝑦)) = (((𝑎‘𝑦)(+g‘(𝑅‘𝑦))(𝑏‘𝑦))(+g‘(𝑅‘𝑦))(𝑐‘𝑦))) |
37 | 3, 4, 20, 21, 23, 27, 29, 5, 25 | prdsplusgfval 16979 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑏(+g‘𝑌)𝑐)‘𝑦) = ((𝑏‘𝑦)(+g‘(𝑅‘𝑦))(𝑐‘𝑦))) |
38 | 37 | oveq2d 7229 |
. . . . 5
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑎‘𝑦)(+g‘(𝑅‘𝑦))((𝑏(+g‘𝑌)𝑐)‘𝑦)) = ((𝑎‘𝑦)(+g‘(𝑅‘𝑦))((𝑏‘𝑦)(+g‘(𝑅‘𝑦))(𝑐‘𝑦)))) |
39 | 34, 36, 38 | 3eqtr4d 2787 |
. . . 4
⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (((𝑎(+g‘𝑌)𝑏)‘𝑦)(+g‘(𝑅‘𝑦))(𝑐‘𝑦)) = ((𝑎‘𝑦)(+g‘(𝑅‘𝑦))((𝑏(+g‘𝑌)𝑐)‘𝑦))) |
40 | 39 | mpteq2dva 5150 |
. . 3
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑦 ∈ 𝐼 ↦ (((𝑎(+g‘𝑌)𝑏)‘𝑦)(+g‘(𝑅‘𝑦))(𝑐‘𝑦))) = (𝑦 ∈ 𝐼 ↦ ((𝑎‘𝑦)(+g‘(𝑅‘𝑦))((𝑏(+g‘𝑌)𝑐)‘𝑦)))) |
41 | 7 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ V) |
42 | 10 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V) |
43 | 22 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼) |
44 | 16 | 3adantr3 1173 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(+g‘𝑌)𝑏) ∈ (Base‘𝑌)) |
45 | | simpr3 1198 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌)) |
46 | 3, 4, 41, 42, 43, 44, 45, 5 | prdsplusgval 16978 |
. . 3
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(+g‘𝑌)𝑏)(+g‘𝑌)𝑐) = (𝑦 ∈ 𝐼 ↦ (((𝑎(+g‘𝑌)𝑏)‘𝑦)(+g‘(𝑅‘𝑦))(𝑐‘𝑦)))) |
47 | | simpr1 1196 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑌)) |
48 | 12 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Mnd) |
49 | | simpr2 1197 |
. . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌)) |
50 | 3, 4, 5, 41, 42, 48, 49, 45 | prdsplusgcl 18204 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑏(+g‘𝑌)𝑐) ∈ (Base‘𝑌)) |
51 | 3, 4, 41, 42, 43, 47, 50, 5 | prdsplusgval 16978 |
. . 3
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(+g‘𝑌)(𝑏(+g‘𝑌)𝑐)) = (𝑦 ∈ 𝐼 ↦ ((𝑎‘𝑦)(+g‘(𝑅‘𝑦))((𝑏(+g‘𝑌)𝑐)‘𝑦)))) |
52 | 40, 46, 51 | 3eqtr4d 2787 |
. 2
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(+g‘𝑌)𝑏)(+g‘𝑌)𝑐) = (𝑎(+g‘𝑌)(𝑏(+g‘𝑌)𝑐))) |
53 | | eqid 2737 |
. . . 4
⊢
(0g ∘ 𝑅) = (0g ∘ 𝑅) |
54 | 3, 4, 5, 7, 10, 12, 53 | prdsidlem 18205 |
. . 3
⊢ (𝜑 → ((0g ∘
𝑅) ∈ (Base‘𝑌) ∧ ∀𝑎 ∈ (Base‘𝑌)(((0g ∘ 𝑅)(+g‘𝑌)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑌)(0g ∘ 𝑅)) = 𝑎))) |
55 | 54 | simpld 498 |
. 2
⊢ (𝜑 → (0g ∘
𝑅) ∈ (Base‘𝑌)) |
56 | 54 | simprd 499 |
. . . 4
⊢ (𝜑 → ∀𝑎 ∈ (Base‘𝑌)(((0g ∘ 𝑅)(+g‘𝑌)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑌)(0g ∘ 𝑅)) = 𝑎)) |
57 | 56 | r19.21bi 3130 |
. . 3
⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → (((0g ∘ 𝑅)(+g‘𝑌)𝑎) = 𝑎 ∧ (𝑎(+g‘𝑌)(0g ∘ 𝑅)) = 𝑎)) |
58 | 57 | simpld 498 |
. 2
⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → ((0g ∘ 𝑅)(+g‘𝑌)𝑎) = 𝑎) |
59 | 57 | simprd 499 |
. 2
⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → (𝑎(+g‘𝑌)(0g ∘ 𝑅)) = 𝑎) |
60 | 1, 2, 17, 52, 55, 58, 59 | ismndd 18195 |
1
⊢ (𝜑 → 𝑌 ∈ Mnd) |