Step | Hyp | Ref
| Expression |
1 | | ernggrp.h-r |
. . . 4
⊢ 𝐻 = (LHyp‘𝐾) |
2 | | ernggrplem.t-r |
. . . 4
⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
3 | | ernggrplem.e-r |
. . . 4
⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
4 | | ernggrp.d-r |
. . . 4
⊢ 𝐷 =
((EDRingR‘𝐾)‘𝑊) |
5 | | eqid 2738 |
. . . 4
⊢
(Base‘𝐷) =
(Base‘𝐷) |
6 | 1, 2, 3, 4, 5 | erngbase-rN 38823 |
. . 3
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸) |
7 | 6 | eqcomd 2744 |
. 2
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐸 = (Base‘𝐷)) |
8 | | ernggrplem.p-r |
. . 3
⊢ 𝑃 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑎‘𝑓) ∘ (𝑏‘𝑓)))) |
9 | | eqid 2738 |
. . . 4
⊢
(+g‘𝐷) = (+g‘𝐷) |
10 | 1, 2, 3, 4, 9 | erngfplus-rN 38824 |
. . 3
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+g‘𝐷) = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑎‘𝑓) ∘ (𝑏‘𝑓))))) |
11 | 8, 10 | eqtr4id 2797 |
. 2
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑃 = (+g‘𝐷)) |
12 | | erngrnglem.m-r |
. . 3
⊢ 𝑀 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑏 ∘ 𝑎)) |
13 | | eqid 2738 |
. . . 4
⊢
(.r‘𝐷) = (.r‘𝐷) |
14 | 1, 2, 3, 4, 13 | erngfmul-rN 38827 |
. . 3
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (.r‘𝐷) = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑏 ∘ 𝑎))) |
15 | 12, 14 | eqtr4id 2797 |
. 2
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑀 = (.r‘𝐷)) |
16 | | ernggrplem.b-r |
. . 3
⊢ 𝐵 = (Base‘𝐾) |
17 | | ernggrplem.o-r |
. . 3
⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
18 | | ernggrplem.i-r |
. . 3
⊢ 𝐼 = (𝑎 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ◡(𝑎‘𝑓))) |
19 | 1, 4, 16, 2, 3, 8,
17, 18 | erngdvlem1-rN 39010 |
. 2
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Grp) |
20 | 15 | oveqd 7292 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑠𝑀𝑡) = (𝑠(.r‘𝐷)𝑡)) |
21 | 20 | 3ad2ant1 1132 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑠𝑀𝑡) = (𝑠(.r‘𝐷)𝑡)) |
22 | 1, 2, 3, 4, 13 | erngmul-rN 38828 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸)) → (𝑠(.r‘𝐷)𝑡) = (𝑡 ∘ 𝑠)) |
23 | 22 | 3impb 1114 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑠(.r‘𝐷)𝑡) = (𝑡 ∘ 𝑠)) |
24 | 21, 23 | eqtrd 2778 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑠𝑀𝑡) = (𝑡 ∘ 𝑠)) |
25 | 1, 3 | tendococl 38786 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑡 ∈ 𝐸 ∧ 𝑠 ∈ 𝐸) → (𝑡 ∘ 𝑠) ∈ 𝐸) |
26 | 25 | 3com23 1125 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑡 ∘ 𝑠) ∈ 𝐸) |
27 | 24, 26 | eqeltrd 2839 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑠𝑀𝑡) ∈ 𝐸) |
28 | 15 | oveqdr 7303 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑡𝑀𝑢) = (𝑡(.r‘𝐷)𝑢)) |
29 | 1, 2, 3, 4, 13 | erngmul-rN 38828 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑡(.r‘𝐷)𝑢) = (𝑢 ∘ 𝑡)) |
30 | 29 | 3adantr1 1168 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑡(.r‘𝐷)𝑢) = (𝑢 ∘ 𝑡)) |
31 | 28, 30 | eqtrd 2778 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑡𝑀𝑢) = (𝑢 ∘ 𝑡)) |
32 | 31 | coeq1d 5770 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑡𝑀𝑢) ∘ 𝑠) = ((𝑢 ∘ 𝑡) ∘ 𝑠)) |
33 | 15 | oveqd 7292 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑠𝑀(𝑡𝑀𝑢)) = (𝑠(.r‘𝐷)(𝑡𝑀𝑢))) |
34 | 33 | adantr 481 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀(𝑡𝑀𝑢)) = (𝑠(.r‘𝐷)(𝑡𝑀𝑢))) |
35 | | simpl 483 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
36 | | simpr1 1193 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → 𝑠 ∈ 𝐸) |
37 | | simpr3 1195 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → 𝑢 ∈ 𝐸) |
38 | | simpr2 1194 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → 𝑡 ∈ 𝐸) |
39 | 1, 3 | tendococl 38786 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑢 ∘ 𝑡) ∈ 𝐸) |
40 | 35, 37, 38, 39 | syl3anc 1370 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑢 ∘ 𝑡) ∈ 𝐸) |
41 | 31, 40 | eqeltrd 2839 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑡𝑀𝑢) ∈ 𝐸) |
42 | 1, 2, 3, 4, 13 | erngmul-rN 38828 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ (𝑡𝑀𝑢) ∈ 𝐸)) → (𝑠(.r‘𝐷)(𝑡𝑀𝑢)) = ((𝑡𝑀𝑢) ∘ 𝑠)) |
43 | 35, 36, 41, 42 | syl12anc 834 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠(.r‘𝐷)(𝑡𝑀𝑢)) = ((𝑡𝑀𝑢) ∘ 𝑠)) |
44 | 34, 43 | eqtrd 2778 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀(𝑡𝑀𝑢)) = ((𝑡𝑀𝑢) ∘ 𝑠)) |
45 | 15 | oveqd 7292 |
. . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((𝑠𝑀𝑡)𝑀𝑢) = ((𝑠𝑀𝑡)(.r‘𝐷)𝑢)) |
46 | 45 | adantr 481 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑀𝑡)𝑀𝑢) = ((𝑠𝑀𝑡)(.r‘𝐷)𝑢)) |
47 | 27 | 3adant3r3 1183 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀𝑡) ∈ 𝐸) |
48 | 1, 2, 3, 4, 13 | erngmul-rN 38828 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑠𝑀𝑡) ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑀𝑡)(.r‘𝐷)𝑢) = (𝑢 ∘ (𝑠𝑀𝑡))) |
49 | 35, 47, 37, 48 | syl12anc 834 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑀𝑡)(.r‘𝐷)𝑢) = (𝑢 ∘ (𝑠𝑀𝑡))) |
50 | 15 | oveqdr 7303 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀𝑡) = (𝑠(.r‘𝐷)𝑡)) |
51 | 22 | 3adantr3 1170 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠(.r‘𝐷)𝑡) = (𝑡 ∘ 𝑠)) |
52 | 50, 51 | eqtrd 2778 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀𝑡) = (𝑡 ∘ 𝑠)) |
53 | 52 | coeq2d 5771 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑢 ∘ (𝑠𝑀𝑡)) = (𝑢 ∘ (𝑡 ∘ 𝑠))) |
54 | 46, 49, 53 | 3eqtrd 2782 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑀𝑡)𝑀𝑢) = (𝑢 ∘ (𝑡 ∘ 𝑠))) |
55 | | coass 6169 |
. . . 4
⊢ ((𝑢 ∘ 𝑡) ∘ 𝑠) = (𝑢 ∘ (𝑡 ∘ 𝑠)) |
56 | 54, 55 | eqtr4di 2796 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑀𝑡)𝑀𝑢) = ((𝑢 ∘ 𝑡) ∘ 𝑠)) |
57 | 32, 44, 56 | 3eqtr4rd 2789 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑀𝑡)𝑀𝑢) = (𝑠𝑀(𝑡𝑀𝑢))) |
58 | 1, 2, 3, 8 | tendodi2 38799 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸 ∧ 𝑠 ∈ 𝐸)) → ((𝑡𝑃𝑢) ∘ 𝑠) = ((𝑡 ∘ 𝑠)𝑃(𝑢 ∘ 𝑠))) |
59 | 35, 38, 37, 36, 58 | syl13anc 1371 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑡𝑃𝑢) ∘ 𝑠) = ((𝑡 ∘ 𝑠)𝑃(𝑢 ∘ 𝑠))) |
60 | 15 | oveqd 7292 |
. . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑠𝑀(𝑡𝑃𝑢)) = (𝑠(.r‘𝐷)(𝑡𝑃𝑢))) |
61 | 60 | adantr 481 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀(𝑡𝑃𝑢)) = (𝑠(.r‘𝐷)(𝑡𝑃𝑢))) |
62 | 1, 2, 3, 8 | tendoplcl 38795 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸) → (𝑡𝑃𝑢) ∈ 𝐸) |
63 | 35, 38, 37, 62 | syl3anc 1370 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑡𝑃𝑢) ∈ 𝐸) |
64 | 1, 2, 3, 4, 13 | erngmul-rN 38828 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ (𝑡𝑃𝑢) ∈ 𝐸)) → (𝑠(.r‘𝐷)(𝑡𝑃𝑢)) = ((𝑡𝑃𝑢) ∘ 𝑠)) |
65 | 35, 36, 63, 64 | syl12anc 834 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠(.r‘𝐷)(𝑡𝑃𝑢)) = ((𝑡𝑃𝑢) ∘ 𝑠)) |
66 | 61, 65 | eqtrd 2778 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀(𝑡𝑃𝑢)) = ((𝑡𝑃𝑢) ∘ 𝑠)) |
67 | 15 | oveqdr 7303 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀𝑢) = (𝑠(.r‘𝐷)𝑢)) |
68 | 1, 2, 3, 4, 13 | erngmul-rN 38828 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠(.r‘𝐷)𝑢) = (𝑢 ∘ 𝑠)) |
69 | 68 | 3adantr2 1169 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠(.r‘𝐷)𝑢) = (𝑢 ∘ 𝑠)) |
70 | 67, 69 | eqtrd 2778 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀𝑢) = (𝑢 ∘ 𝑠)) |
71 | 52, 70 | oveq12d 7293 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑀𝑡)𝑃(𝑠𝑀𝑢)) = ((𝑡 ∘ 𝑠)𝑃(𝑢 ∘ 𝑠))) |
72 | 59, 66, 71 | 3eqtr4d 2788 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀(𝑡𝑃𝑢)) = ((𝑠𝑀𝑡)𝑃(𝑠𝑀𝑢))) |
73 | 1, 2, 3, 8 | tendodi1 38798 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑢 ∈ 𝐸 ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸)) → (𝑢 ∘ (𝑠𝑃𝑡)) = ((𝑢 ∘ 𝑠)𝑃(𝑢 ∘ 𝑡))) |
74 | 35, 37, 36, 38, 73 | syl13anc 1371 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑢 ∘ (𝑠𝑃𝑡)) = ((𝑢 ∘ 𝑠)𝑃(𝑢 ∘ 𝑡))) |
75 | 15 | adantr 481 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → 𝑀 = (.r‘𝐷)) |
76 | 75 | oveqd 7292 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑃𝑡)𝑀𝑢) = ((𝑠𝑃𝑡)(.r‘𝐷)𝑢)) |
77 | 1, 2, 3, 8 | tendoplcl 38795 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑠𝑃𝑡) ∈ 𝐸) |
78 | 77 | 3adant3r3 1183 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑃𝑡) ∈ 𝐸) |
79 | 1, 2, 3, 4, 13 | erngmul-rN 38828 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑠𝑃𝑡) ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑃𝑡)(.r‘𝐷)𝑢) = (𝑢 ∘ (𝑠𝑃𝑡))) |
80 | 35, 78, 37, 79 | syl12anc 834 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑃𝑡)(.r‘𝐷)𝑢) = (𝑢 ∘ (𝑠𝑃𝑡))) |
81 | 76, 80 | eqtrd 2778 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑃𝑡)𝑀𝑢) = (𝑢 ∘ (𝑠𝑃𝑡))) |
82 | 70, 31 | oveq12d 7293 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑀𝑢)𝑃(𝑡𝑀𝑢)) = ((𝑢 ∘ 𝑠)𝑃(𝑢 ∘ 𝑡))) |
83 | 74, 81, 82 | 3eqtr4d 2788 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑃𝑡)𝑀𝑢) = ((𝑠𝑀𝑢)𝑃(𝑡𝑀𝑢))) |
84 | 1, 2, 3 | tendoidcl 38783 |
. 2
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ 𝐸) |
85 | 15 | oveqd 7292 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (( I ↾ 𝑇)𝑀𝑠) = (( I ↾ 𝑇)(.r‘𝐷)𝑠)) |
86 | 85 | adantr 481 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (( I ↾ 𝑇)𝑀𝑠) = (( I ↾ 𝑇)(.r‘𝐷)𝑠)) |
87 | | simpl 483 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
88 | 84 | adantr 481 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → ( I ↾ 𝑇) ∈ 𝐸) |
89 | | simpr 485 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → 𝑠 ∈ 𝐸) |
90 | 1, 2, 3, 4, 13 | erngmul-rN 38828 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (( I ↾ 𝑇) ∈ 𝐸 ∧ 𝑠 ∈ 𝐸)) → (( I ↾ 𝑇)(.r‘𝐷)𝑠) = (𝑠 ∘ ( I ↾ 𝑇))) |
91 | 87, 88, 89, 90 | syl12anc 834 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (( I ↾ 𝑇)(.r‘𝐷)𝑠) = (𝑠 ∘ ( I ↾ 𝑇))) |
92 | 1, 2, 3 | tendo1mulr 38785 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (𝑠 ∘ ( I ↾ 𝑇)) = 𝑠) |
93 | 86, 91, 92 | 3eqtrd 2782 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (( I ↾ 𝑇)𝑀𝑠) = 𝑠) |
94 | 15 | oveqd 7292 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑠𝑀( I ↾ 𝑇)) = (𝑠(.r‘𝐷)( I ↾ 𝑇))) |
95 | 94 | adantr 481 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (𝑠𝑀( I ↾ 𝑇)) = (𝑠(.r‘𝐷)( I ↾ 𝑇))) |
96 | 1, 2, 3, 4, 13 | erngmul-rN 38828 |
. . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ ( I ↾ 𝑇) ∈ 𝐸)) → (𝑠(.r‘𝐷)( I ↾ 𝑇)) = (( I ↾ 𝑇) ∘ 𝑠)) |
97 | 87, 89, 88, 96 | syl12anc 834 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (𝑠(.r‘𝐷)( I ↾ 𝑇)) = (( I ↾ 𝑇) ∘ 𝑠)) |
98 | 1, 2, 3 | tendo1mul 38784 |
. . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (( I ↾ 𝑇) ∘ 𝑠) = 𝑠) |
99 | 95, 97, 98 | 3eqtrd 2782 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (𝑠𝑀( I ↾ 𝑇)) = 𝑠) |
100 | 7, 11, 15, 19, 27, 57, 72, 83, 84, 93, 99 | isringd 19824 |
1
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Ring) |