| 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 2736 | . . . 4
⊢
(Base‘𝐷) =
(Base‘𝐷) | 
| 6 | 1, 2, 3, 4, 5 | erngbase-rN 40812 | . . 3
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸) | 
| 7 | 6 | eqcomd 2742 | . 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 2736 | . . . 4
⊢
(+g‘𝐷) = (+g‘𝐷) | 
| 10 | 1, 2, 3, 4, 9 | erngfplus-rN 40813 | . . 3
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+g‘𝐷) = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑎‘𝑓) ∘ (𝑏‘𝑓))))) | 
| 11 | 8, 10 | eqtr4id 2795 | . 2
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑃 = (+g‘𝐷)) | 
| 12 |  | erngrnglem.m-r | . . 3
⊢ 𝑀 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑏 ∘ 𝑎)) | 
| 13 |  | eqid 2736 | . . . 4
⊢
(.r‘𝐷) = (.r‘𝐷) | 
| 14 | 1, 2, 3, 4, 13 | erngfmul-rN 40816 | . . 3
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (.r‘𝐷) = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑏 ∘ 𝑎))) | 
| 15 | 12, 14 | eqtr4id 2795 | . 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 40999 | . 2
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Grp) | 
| 20 | 15 | oveqd 7449 | . . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑠𝑀𝑡) = (𝑠(.r‘𝐷)𝑡)) | 
| 21 | 20 | 3ad2ant1 1133 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑠𝑀𝑡) = (𝑠(.r‘𝐷)𝑡)) | 
| 22 | 1, 2, 3, 4, 13 | erngmul-rN 40817 | . . . . 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 2776 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑠𝑀𝑡) = (𝑡 ∘ 𝑠)) | 
| 25 | 1, 3 | tendococl 40775 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑡 ∈ 𝐸 ∧ 𝑠 ∈ 𝐸) → (𝑡 ∘ 𝑠) ∈ 𝐸) | 
| 26 | 25 | 3com23 1126 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑡 ∘ 𝑠) ∈ 𝐸) | 
| 27 | 24, 26 | eqeltrd 2840 | . 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑠𝑀𝑡) ∈ 𝐸) | 
| 28 | 15 | oveqdr 7460 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑡𝑀𝑢) = (𝑡(.r‘𝐷)𝑢)) | 
| 29 | 1, 2, 3, 4, 13 | erngmul-rN 40817 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑡(.r‘𝐷)𝑢) = (𝑢 ∘ 𝑡)) | 
| 30 | 29 | 3adantr1 1169 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑡(.r‘𝐷)𝑢) = (𝑢 ∘ 𝑡)) | 
| 31 | 28, 30 | eqtrd 2776 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑡𝑀𝑢) = (𝑢 ∘ 𝑡)) | 
| 32 | 31 | coeq1d 5871 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑡𝑀𝑢) ∘ 𝑠) = ((𝑢 ∘ 𝑡) ∘ 𝑠)) | 
| 33 | 15 | oveqd 7449 | . . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑠𝑀(𝑡𝑀𝑢)) = (𝑠(.r‘𝐷)(𝑡𝑀𝑢))) | 
| 34 | 33 | adantr 480 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀(𝑡𝑀𝑢)) = (𝑠(.r‘𝐷)(𝑡𝑀𝑢))) | 
| 35 |  | simpl 482 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | 
| 36 |  | simpr1 1194 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → 𝑠 ∈ 𝐸) | 
| 37 |  | simpr3 1196 | . . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → 𝑢 ∈ 𝐸) | 
| 38 |  | simpr2 1195 | . . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → 𝑡 ∈ 𝐸) | 
| 39 | 1, 3 | tendococl 40775 | . . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑢 ∘ 𝑡) ∈ 𝐸) | 
| 40 | 35, 37, 38, 39 | syl3anc 1372 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑢 ∘ 𝑡) ∈ 𝐸) | 
| 41 | 31, 40 | eqeltrd 2840 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑡𝑀𝑢) ∈ 𝐸) | 
| 42 | 1, 2, 3, 4, 13 | erngmul-rN 40817 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ (𝑡𝑀𝑢) ∈ 𝐸)) → (𝑠(.r‘𝐷)(𝑡𝑀𝑢)) = ((𝑡𝑀𝑢) ∘ 𝑠)) | 
| 43 | 35, 36, 41, 42 | syl12anc 836 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠(.r‘𝐷)(𝑡𝑀𝑢)) = ((𝑡𝑀𝑢) ∘ 𝑠)) | 
| 44 | 34, 43 | eqtrd 2776 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀(𝑡𝑀𝑢)) = ((𝑡𝑀𝑢) ∘ 𝑠)) | 
| 45 | 15 | oveqd 7449 | . . . . . 6
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((𝑠𝑀𝑡)𝑀𝑢) = ((𝑠𝑀𝑡)(.r‘𝐷)𝑢)) | 
| 46 | 45 | adantr 480 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑀𝑡)𝑀𝑢) = ((𝑠𝑀𝑡)(.r‘𝐷)𝑢)) | 
| 47 | 27 | 3adant3r3 1184 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀𝑡) ∈ 𝐸) | 
| 48 | 1, 2, 3, 4, 13 | erngmul-rN 40817 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑠𝑀𝑡) ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑀𝑡)(.r‘𝐷)𝑢) = (𝑢 ∘ (𝑠𝑀𝑡))) | 
| 49 | 35, 47, 37, 48 | syl12anc 836 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑀𝑡)(.r‘𝐷)𝑢) = (𝑢 ∘ (𝑠𝑀𝑡))) | 
| 50 | 15 | oveqdr 7460 | . . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀𝑡) = (𝑠(.r‘𝐷)𝑡)) | 
| 51 | 22 | 3adantr3 1171 | . . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠(.r‘𝐷)𝑡) = (𝑡 ∘ 𝑠)) | 
| 52 | 50, 51 | eqtrd 2776 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀𝑡) = (𝑡 ∘ 𝑠)) | 
| 53 | 52 | coeq2d 5872 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑢 ∘ (𝑠𝑀𝑡)) = (𝑢 ∘ (𝑡 ∘ 𝑠))) | 
| 54 | 46, 49, 53 | 3eqtrd 2780 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑀𝑡)𝑀𝑢) = (𝑢 ∘ (𝑡 ∘ 𝑠))) | 
| 55 |  | coass 6284 | . . . 4
⊢ ((𝑢 ∘ 𝑡) ∘ 𝑠) = (𝑢 ∘ (𝑡 ∘ 𝑠)) | 
| 56 | 54, 55 | eqtr4di 2794 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑀𝑡)𝑀𝑢) = ((𝑢 ∘ 𝑡) ∘ 𝑠)) | 
| 57 | 32, 44, 56 | 3eqtr4rd 2787 | . 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑀𝑡)𝑀𝑢) = (𝑠𝑀(𝑡𝑀𝑢))) | 
| 58 | 1, 2, 3, 8 | tendodi2 40788 | . . . 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 1373 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑡𝑃𝑢) ∘ 𝑠) = ((𝑡 ∘ 𝑠)𝑃(𝑢 ∘ 𝑠))) | 
| 60 | 15 | oveqd 7449 | . . . . 5
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑠𝑀(𝑡𝑃𝑢)) = (𝑠(.r‘𝐷)(𝑡𝑃𝑢))) | 
| 61 | 60 | adantr 480 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀(𝑡𝑃𝑢)) = (𝑠(.r‘𝐷)(𝑡𝑃𝑢))) | 
| 62 | 1, 2, 3, 8 | tendoplcl 40784 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸) → (𝑡𝑃𝑢) ∈ 𝐸) | 
| 63 | 35, 38, 37, 62 | syl3anc 1372 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑡𝑃𝑢) ∈ 𝐸) | 
| 64 | 1, 2, 3, 4, 13 | erngmul-rN 40817 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ (𝑡𝑃𝑢) ∈ 𝐸)) → (𝑠(.r‘𝐷)(𝑡𝑃𝑢)) = ((𝑡𝑃𝑢) ∘ 𝑠)) | 
| 65 | 35, 36, 63, 64 | syl12anc 836 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠(.r‘𝐷)(𝑡𝑃𝑢)) = ((𝑡𝑃𝑢) ∘ 𝑠)) | 
| 66 | 61, 65 | eqtrd 2776 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀(𝑡𝑃𝑢)) = ((𝑡𝑃𝑢) ∘ 𝑠)) | 
| 67 | 15 | oveqdr 7460 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀𝑢) = (𝑠(.r‘𝐷)𝑢)) | 
| 68 | 1, 2, 3, 4, 13 | erngmul-rN 40817 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠(.r‘𝐷)𝑢) = (𝑢 ∘ 𝑠)) | 
| 69 | 68 | 3adantr2 1170 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠(.r‘𝐷)𝑢) = (𝑢 ∘ 𝑠)) | 
| 70 | 67, 69 | eqtrd 2776 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀𝑢) = (𝑢 ∘ 𝑠)) | 
| 71 | 52, 70 | oveq12d 7450 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑀𝑡)𝑃(𝑠𝑀𝑢)) = ((𝑡 ∘ 𝑠)𝑃(𝑢 ∘ 𝑠))) | 
| 72 | 59, 66, 71 | 3eqtr4d 2786 | . 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀(𝑡𝑃𝑢)) = ((𝑠𝑀𝑡)𝑃(𝑠𝑀𝑢))) | 
| 73 | 1, 2, 3, 8 | tendodi1 40787 | . . . 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 1373 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑢 ∘ (𝑠𝑃𝑡)) = ((𝑢 ∘ 𝑠)𝑃(𝑢 ∘ 𝑡))) | 
| 75 | 15 | adantr 480 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → 𝑀 = (.r‘𝐷)) | 
| 76 | 75 | oveqd 7449 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑃𝑡)𝑀𝑢) = ((𝑠𝑃𝑡)(.r‘𝐷)𝑢)) | 
| 77 | 1, 2, 3, 8 | tendoplcl 40784 | . . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑠𝑃𝑡) ∈ 𝐸) | 
| 78 | 77 | 3adant3r3 1184 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑃𝑡) ∈ 𝐸) | 
| 79 | 1, 2, 3, 4, 13 | erngmul-rN 40817 | . . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑠𝑃𝑡) ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑃𝑡)(.r‘𝐷)𝑢) = (𝑢 ∘ (𝑠𝑃𝑡))) | 
| 80 | 35, 78, 37, 79 | syl12anc 836 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑃𝑡)(.r‘𝐷)𝑢) = (𝑢 ∘ (𝑠𝑃𝑡))) | 
| 81 | 76, 80 | eqtrd 2776 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑃𝑡)𝑀𝑢) = (𝑢 ∘ (𝑠𝑃𝑡))) | 
| 82 | 70, 31 | oveq12d 7450 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑀𝑢)𝑃(𝑡𝑀𝑢)) = ((𝑢 ∘ 𝑠)𝑃(𝑢 ∘ 𝑡))) | 
| 83 | 74, 81, 82 | 3eqtr4d 2786 | . 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑃𝑡)𝑀𝑢) = ((𝑠𝑀𝑢)𝑃(𝑡𝑀𝑢))) | 
| 84 | 1, 2, 3 | tendoidcl 40772 | . 2
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ 𝐸) | 
| 85 | 15 | oveqd 7449 | . . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (( I ↾ 𝑇)𝑀𝑠) = (( I ↾ 𝑇)(.r‘𝐷)𝑠)) | 
| 86 | 85 | adantr 480 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (( I ↾ 𝑇)𝑀𝑠) = (( I ↾ 𝑇)(.r‘𝐷)𝑠)) | 
| 87 |  | simpl 482 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | 
| 88 | 84 | adantr 480 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → ( I ↾ 𝑇) ∈ 𝐸) | 
| 89 |  | simpr 484 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → 𝑠 ∈ 𝐸) | 
| 90 | 1, 2, 3, 4, 13 | erngmul-rN 40817 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (( I ↾ 𝑇) ∈ 𝐸 ∧ 𝑠 ∈ 𝐸)) → (( I ↾ 𝑇)(.r‘𝐷)𝑠) = (𝑠 ∘ ( I ↾ 𝑇))) | 
| 91 | 87, 88, 89, 90 | syl12anc 836 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (( I ↾ 𝑇)(.r‘𝐷)𝑠) = (𝑠 ∘ ( I ↾ 𝑇))) | 
| 92 | 1, 2, 3 | tendo1mulr 40774 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (𝑠 ∘ ( I ↾ 𝑇)) = 𝑠) | 
| 93 | 86, 91, 92 | 3eqtrd 2780 | . 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (( I ↾ 𝑇)𝑀𝑠) = 𝑠) | 
| 94 | 15 | oveqd 7449 | . . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑠𝑀( I ↾ 𝑇)) = (𝑠(.r‘𝐷)( I ↾ 𝑇))) | 
| 95 | 94 | adantr 480 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (𝑠𝑀( I ↾ 𝑇)) = (𝑠(.r‘𝐷)( I ↾ 𝑇))) | 
| 96 | 1, 2, 3, 4, 13 | erngmul-rN 40817 | . . . 4
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ ( I ↾ 𝑇) ∈ 𝐸)) → (𝑠(.r‘𝐷)( I ↾ 𝑇)) = (( I ↾ 𝑇) ∘ 𝑠)) | 
| 97 | 87, 89, 88, 96 | syl12anc 836 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (𝑠(.r‘𝐷)( I ↾ 𝑇)) = (( I ↾ 𝑇) ∘ 𝑠)) | 
| 98 | 1, 2, 3 | tendo1mul 40773 | . . 3
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (( I ↾ 𝑇) ∘ 𝑠) = 𝑠) | 
| 99 | 95, 97, 98 | 3eqtrd 2780 | . 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (𝑠𝑀( I ↾ 𝑇)) = 𝑠) | 
| 100 | 7, 11, 15, 19, 27, 57, 72, 83, 84, 93, 99 | isringd 20289 | 1
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Ring) |