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Theorem dvhlveclem 40071
Description: Lemma for dvhlvec 40072. TODO: proof substituting inner part first shorter/longer than substituting outer part first? TODO: break up into smaller lemmas? TODO: does πœ‘ β†’ method shorten proof? (Contributed by NM, 22-Oct-2013.) (Proof shortened by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
dvhgrp.b 𝐡 = (Baseβ€˜πΎ)
dvhgrp.h 𝐻 = (LHypβ€˜πΎ)
dvhgrp.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
dvhgrp.e 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
dvhgrp.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
dvhgrp.d 𝐷 = (Scalarβ€˜π‘ˆ)
dvhgrp.p ⨣ = (+gβ€˜π·)
dvhgrp.a + = (+gβ€˜π‘ˆ)
dvhgrp.o 0 = (0gβ€˜π·)
dvhgrp.i 𝐼 = (invgβ€˜π·)
dvhlvec.m Γ— = (.rβ€˜π·)
dvhlvec.s Β· = ( ·𝑠 β€˜π‘ˆ)
Assertion
Ref Expression
dvhlveclem ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘ˆ ∈ LVec)

Proof of Theorem dvhlveclem
Dummy variables 𝑑 𝑓 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvhgrp.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
2 dvhgrp.t . . . . 5 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
3 dvhgrp.e . . . . 5 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
4 dvhgrp.u . . . . 5 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
5 eqid 2732 . . . . 5 (Baseβ€˜π‘ˆ) = (Baseβ€˜π‘ˆ)
61, 2, 3, 4, 5dvhvbase 40050 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (Baseβ€˜π‘ˆ) = (𝑇 Γ— 𝐸))
76eqcomd 2738 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (𝑇 Γ— 𝐸) = (Baseβ€˜π‘ˆ))
8 dvhgrp.a . . . 4 + = (+gβ€˜π‘ˆ)
98a1i 11 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ + = (+gβ€˜π‘ˆ))
10 dvhgrp.d . . . 4 𝐷 = (Scalarβ€˜π‘ˆ)
1110a1i 11 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 = (Scalarβ€˜π‘ˆ))
12 dvhlvec.s . . . 4 Β· = ( ·𝑠 β€˜π‘ˆ)
1312a1i 11 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ Β· = ( ·𝑠 β€˜π‘ˆ))
14 eqid 2732 . . . . 5 (Baseβ€˜π·) = (Baseβ€˜π·)
151, 3, 4, 10, 14dvhbase 40046 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (Baseβ€˜π·) = 𝐸)
1615eqcomd 2738 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐸 = (Baseβ€˜π·))
17 dvhgrp.p . . . 4 ⨣ = (+gβ€˜π·)
1817a1i 11 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ⨣ = (+gβ€˜π·))
19 dvhlvec.m . . . 4 Γ— = (.rβ€˜π·)
2019a1i 11 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ Γ— = (.rβ€˜π·))
21 eqid 2732 . . . . . 6 ((EDRingβ€˜πΎ)β€˜π‘Š) = ((EDRingβ€˜πΎ)β€˜π‘Š)
221, 21, 4, 10dvhsca 40045 . . . . 5 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 = ((EDRingβ€˜πΎ)β€˜π‘Š))
2322fveq2d 6895 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (1rβ€˜π·) = (1rβ€˜((EDRingβ€˜πΎ)β€˜π‘Š)))
24 eqid 2732 . . . . 5 (1rβ€˜((EDRingβ€˜πΎ)β€˜π‘Š)) = (1rβ€˜((EDRingβ€˜πΎ)β€˜π‘Š))
251, 2, 21, 24erng1r 39958 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (1rβ€˜((EDRingβ€˜πΎ)β€˜π‘Š)) = ( I β†Ύ 𝑇))
2623, 25eqtr2d 2773 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ( I β†Ύ 𝑇) = (1rβ€˜π·))
271, 21erngdv 39956 . . . . 5 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ((EDRingβ€˜πΎ)β€˜π‘Š) ∈ DivRing)
2822, 27eqeltrd 2833 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 ∈ DivRing)
29 drngring 20368 . . . 4 (𝐷 ∈ DivRing β†’ 𝐷 ∈ Ring)
3028, 29syl 17 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 ∈ Ring)
31 dvhgrp.b . . . 4 𝐡 = (Baseβ€˜πΎ)
32 dvhgrp.o . . . 4 0 = (0gβ€˜π·)
33 dvhgrp.i . . . 4 𝐼 = (invgβ€˜π·)
3431, 1, 2, 3, 4, 10, 17, 8, 32, 33dvhgrp 40070 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘ˆ ∈ Grp)
351, 2, 3, 4, 12dvhvscacl 40066 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 Β· 𝑑) ∈ (𝑇 Γ— 𝐸))
36353impb 1115 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸)) β†’ (𝑠 Β· 𝑑) ∈ (𝑇 Γ— 𝐸))
37 simpl 483 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
38 simpr1 1194 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ 𝑠 ∈ 𝐸)
39 simpr2 1195 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ 𝑑 ∈ (𝑇 Γ— 𝐸))
40 xp1st 8009 . . . . . . . 8 (𝑑 ∈ (𝑇 Γ— 𝐸) β†’ (1st β€˜π‘‘) ∈ 𝑇)
4139, 40syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (1st β€˜π‘‘) ∈ 𝑇)
42 simpr3 1196 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ 𝑓 ∈ (𝑇 Γ— 𝐸))
43 xp1st 8009 . . . . . . . 8 (𝑓 ∈ (𝑇 Γ— 𝐸) β†’ (1st β€˜π‘“) ∈ 𝑇)
4442, 43syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (1st β€˜π‘“) ∈ 𝑇)
451, 2, 3tendospdi1 39983 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ (1st β€˜π‘‘) ∈ 𝑇 ∧ (1st β€˜π‘“) ∈ 𝑇)) β†’ (π‘ β€˜((1st β€˜π‘‘) ∘ (1st β€˜π‘“))) = ((π‘ β€˜(1st β€˜π‘‘)) ∘ (π‘ β€˜(1st β€˜π‘“))))
4637, 38, 41, 44, 45syl13anc 1372 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (π‘ β€˜((1st β€˜π‘‘) ∘ (1st β€˜π‘“))) = ((π‘ β€˜(1st β€˜π‘‘)) ∘ (π‘ β€˜(1st β€˜π‘“))))
471, 2, 3, 4, 10, 8, 17dvhvadd 40055 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑑 + 𝑓) = ⟨((1st β€˜π‘‘) ∘ (1st β€˜π‘“)), ((2nd β€˜π‘‘) ⨣ (2nd β€˜π‘“))⟩)
48473adantr1 1169 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑑 + 𝑓) = ⟨((1st β€˜π‘‘) ∘ (1st β€˜π‘“)), ((2nd β€˜π‘‘) ⨣ (2nd β€˜π‘“))⟩)
4948fveq2d 6895 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (1st β€˜(𝑑 + 𝑓)) = (1st β€˜βŸ¨((1st β€˜π‘‘) ∘ (1st β€˜π‘“)), ((2nd β€˜π‘‘) ⨣ (2nd β€˜π‘“))⟩))
50 fvex 6904 . . . . . . . . . 10 (1st β€˜π‘‘) ∈ V
51 fvex 6904 . . . . . . . . . 10 (1st β€˜π‘“) ∈ V
5250, 51coex 7923 . . . . . . . . 9 ((1st β€˜π‘‘) ∘ (1st β€˜π‘“)) ∈ V
53 ovex 7444 . . . . . . . . 9 ((2nd β€˜π‘‘) ⨣ (2nd β€˜π‘“)) ∈ V
5452, 53op1st 7985 . . . . . . . 8 (1st β€˜βŸ¨((1st β€˜π‘‘) ∘ (1st β€˜π‘“)), ((2nd β€˜π‘‘) ⨣ (2nd β€˜π‘“))⟩) = ((1st β€˜π‘‘) ∘ (1st β€˜π‘“))
5549, 54eqtrdi 2788 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (1st β€˜(𝑑 + 𝑓)) = ((1st β€˜π‘‘) ∘ (1st β€˜π‘“)))
5655fveq2d 6895 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (π‘ β€˜(1st β€˜(𝑑 + 𝑓))) = (π‘ β€˜((1st β€˜π‘‘) ∘ (1st β€˜π‘“))))
571, 2, 3, 4, 12dvhvsca 40064 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 Β· 𝑑) = ⟨(π‘ β€˜(1st β€˜π‘‘)), (𝑠 ∘ (2nd β€˜π‘‘))⟩)
58573adantr3 1171 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 Β· 𝑑) = ⟨(π‘ β€˜(1st β€˜π‘‘)), (𝑠 ∘ (2nd β€˜π‘‘))⟩)
5958fveq2d 6895 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (1st β€˜(𝑠 Β· 𝑑)) = (1st β€˜βŸ¨(π‘ β€˜(1st β€˜π‘‘)), (𝑠 ∘ (2nd β€˜π‘‘))⟩))
60 fvex 6904 . . . . . . . . 9 (π‘ β€˜(1st β€˜π‘‘)) ∈ V
61 vex 3478 . . . . . . . . . 10 𝑠 ∈ V
62 fvex 6904 . . . . . . . . . 10 (2nd β€˜π‘‘) ∈ V
6361, 62coex 7923 . . . . . . . . 9 (𝑠 ∘ (2nd β€˜π‘‘)) ∈ V
6460, 63op1st 7985 . . . . . . . 8 (1st β€˜βŸ¨(π‘ β€˜(1st β€˜π‘‘)), (𝑠 ∘ (2nd β€˜π‘‘))⟩) = (π‘ β€˜(1st β€˜π‘‘))
6559, 64eqtrdi 2788 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (1st β€˜(𝑠 Β· 𝑑)) = (π‘ β€˜(1st β€˜π‘‘)))
661, 2, 3, 4, 12dvhvsca 40064 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 Β· 𝑓) = ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)
67663adantr2 1170 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 Β· 𝑓) = ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)
6867fveq2d 6895 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (1st β€˜(𝑠 Β· 𝑓)) = (1st β€˜βŸ¨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩))
69 fvex 6904 . . . . . . . . 9 (π‘ β€˜(1st β€˜π‘“)) ∈ V
70 fvex 6904 . . . . . . . . . 10 (2nd β€˜π‘“) ∈ V
7161, 70coex 7923 . . . . . . . . 9 (𝑠 ∘ (2nd β€˜π‘“)) ∈ V
7269, 71op1st 7985 . . . . . . . 8 (1st β€˜βŸ¨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩) = (π‘ β€˜(1st β€˜π‘“))
7368, 72eqtrdi 2788 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (1st β€˜(𝑠 Β· 𝑓)) = (π‘ β€˜(1st β€˜π‘“)))
7465, 73coeq12d 5864 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((1st β€˜(𝑠 Β· 𝑑)) ∘ (1st β€˜(𝑠 Β· 𝑓))) = ((π‘ β€˜(1st β€˜π‘‘)) ∘ (π‘ β€˜(1st β€˜π‘“))))
7546, 56, 743eqtr4d 2782 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (π‘ β€˜(1st β€˜(𝑑 + 𝑓))) = ((1st β€˜(𝑠 Β· 𝑑)) ∘ (1st β€˜(𝑠 Β· 𝑓))))
7630adantr 481 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ 𝐷 ∈ Ring)
7716adantr 481 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ 𝐸 = (Baseβ€˜π·))
7838, 77eleqtrd 2835 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ 𝑠 ∈ (Baseβ€˜π·))
79 xp2nd 8010 . . . . . . . . . 10 (𝑑 ∈ (𝑇 Γ— 𝐸) β†’ (2nd β€˜π‘‘) ∈ 𝐸)
8039, 79syl 17 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (2nd β€˜π‘‘) ∈ 𝐸)
8180, 77eleqtrd 2835 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (2nd β€˜π‘‘) ∈ (Baseβ€˜π·))
82 xp2nd 8010 . . . . . . . . . 10 (𝑓 ∈ (𝑇 Γ— 𝐸) β†’ (2nd β€˜π‘“) ∈ 𝐸)
8342, 82syl 17 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (2nd β€˜π‘“) ∈ 𝐸)
8483, 77eleqtrd 2835 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (2nd β€˜π‘“) ∈ (Baseβ€˜π·))
8514, 17, 19ringdi 20083 . . . . . . . 8 ((𝐷 ∈ Ring ∧ (𝑠 ∈ (Baseβ€˜π·) ∧ (2nd β€˜π‘‘) ∈ (Baseβ€˜π·) ∧ (2nd β€˜π‘“) ∈ (Baseβ€˜π·))) β†’ (𝑠 Γ— ((2nd β€˜π‘‘) ⨣ (2nd β€˜π‘“))) = ((𝑠 Γ— (2nd β€˜π‘‘)) ⨣ (𝑠 Γ— (2nd β€˜π‘“))))
8676, 78, 81, 84, 85syl13anc 1372 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 Γ— ((2nd β€˜π‘‘) ⨣ (2nd β€˜π‘“))) = ((𝑠 Γ— (2nd β€˜π‘‘)) ⨣ (𝑠 Γ— (2nd β€˜π‘“))))
8714, 17ringacl 20097 . . . . . . . . . 10 ((𝐷 ∈ Ring ∧ (2nd β€˜π‘‘) ∈ (Baseβ€˜π·) ∧ (2nd β€˜π‘“) ∈ (Baseβ€˜π·)) β†’ ((2nd β€˜π‘‘) ⨣ (2nd β€˜π‘“)) ∈ (Baseβ€˜π·))
8876, 81, 84, 87syl3anc 1371 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((2nd β€˜π‘‘) ⨣ (2nd β€˜π‘“)) ∈ (Baseβ€˜π·))
8988, 77eleqtrrd 2836 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((2nd β€˜π‘‘) ⨣ (2nd β€˜π‘“)) ∈ 𝐸)
901, 2, 3, 4, 10, 19dvhmulr 40049 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ ((2nd β€˜π‘‘) ⨣ (2nd β€˜π‘“)) ∈ 𝐸)) β†’ (𝑠 Γ— ((2nd β€˜π‘‘) ⨣ (2nd β€˜π‘“))) = (𝑠 ∘ ((2nd β€˜π‘‘) ⨣ (2nd β€˜π‘“))))
9137, 38, 89, 90syl12anc 835 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 Γ— ((2nd β€˜π‘‘) ⨣ (2nd β€˜π‘“))) = (𝑠 ∘ ((2nd β€˜π‘‘) ⨣ (2nd β€˜π‘“))))
921, 2, 3, 4, 10, 19dvhmulr 40049 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ (2nd β€˜π‘‘) ∈ 𝐸)) β†’ (𝑠 Γ— (2nd β€˜π‘‘)) = (𝑠 ∘ (2nd β€˜π‘‘)))
9337, 38, 80, 92syl12anc 835 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 Γ— (2nd β€˜π‘‘)) = (𝑠 ∘ (2nd β€˜π‘‘)))
941, 2, 3, 4, 10, 19dvhmulr 40049 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ (2nd β€˜π‘“) ∈ 𝐸)) β†’ (𝑠 Γ— (2nd β€˜π‘“)) = (𝑠 ∘ (2nd β€˜π‘“)))
9537, 38, 83, 94syl12anc 835 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 Γ— (2nd β€˜π‘“)) = (𝑠 ∘ (2nd β€˜π‘“)))
9693, 95oveq12d 7429 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((𝑠 Γ— (2nd β€˜π‘‘)) ⨣ (𝑠 Γ— (2nd β€˜π‘“))) = ((𝑠 ∘ (2nd β€˜π‘‘)) ⨣ (𝑠 ∘ (2nd β€˜π‘“))))
9786, 91, 963eqtr3d 2780 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 ∘ ((2nd β€˜π‘‘) ⨣ (2nd β€˜π‘“))) = ((𝑠 ∘ (2nd β€˜π‘‘)) ⨣ (𝑠 ∘ (2nd β€˜π‘“))))
9848fveq2d 6895 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (2nd β€˜(𝑑 + 𝑓)) = (2nd β€˜βŸ¨((1st β€˜π‘‘) ∘ (1st β€˜π‘“)), ((2nd β€˜π‘‘) ⨣ (2nd β€˜π‘“))⟩))
9952, 53op2nd 7986 . . . . . . . 8 (2nd β€˜βŸ¨((1st β€˜π‘‘) ∘ (1st β€˜π‘“)), ((2nd β€˜π‘‘) ⨣ (2nd β€˜π‘“))⟩) = ((2nd β€˜π‘‘) ⨣ (2nd β€˜π‘“))
10098, 99eqtrdi 2788 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (2nd β€˜(𝑑 + 𝑓)) = ((2nd β€˜π‘‘) ⨣ (2nd β€˜π‘“)))
101100coeq2d 5862 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 ∘ (2nd β€˜(𝑑 + 𝑓))) = (𝑠 ∘ ((2nd β€˜π‘‘) ⨣ (2nd β€˜π‘“))))
10258fveq2d 6895 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (2nd β€˜(𝑠 Β· 𝑑)) = (2nd β€˜βŸ¨(π‘ β€˜(1st β€˜π‘‘)), (𝑠 ∘ (2nd β€˜π‘‘))⟩))
10360, 63op2nd 7986 . . . . . . . 8 (2nd β€˜βŸ¨(π‘ β€˜(1st β€˜π‘‘)), (𝑠 ∘ (2nd β€˜π‘‘))⟩) = (𝑠 ∘ (2nd β€˜π‘‘))
104102, 103eqtrdi 2788 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (2nd β€˜(𝑠 Β· 𝑑)) = (𝑠 ∘ (2nd β€˜π‘‘)))
10567fveq2d 6895 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (2nd β€˜(𝑠 Β· 𝑓)) = (2nd β€˜βŸ¨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩))
10669, 71op2nd 7986 . . . . . . . 8 (2nd β€˜βŸ¨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩) = (𝑠 ∘ (2nd β€˜π‘“))
107105, 106eqtrdi 2788 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (2nd β€˜(𝑠 Β· 𝑓)) = (𝑠 ∘ (2nd β€˜π‘“)))
108104, 107oveq12d 7429 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((2nd β€˜(𝑠 Β· 𝑑)) ⨣ (2nd β€˜(𝑠 Β· 𝑓))) = ((𝑠 ∘ (2nd β€˜π‘‘)) ⨣ (𝑠 ∘ (2nd β€˜π‘“))))
10997, 101, 1083eqtr4d 2782 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 ∘ (2nd β€˜(𝑑 + 𝑓))) = ((2nd β€˜(𝑠 Β· 𝑑)) ⨣ (2nd β€˜(𝑠 Β· 𝑓))))
11075, 109opeq12d 4881 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ⟨(π‘ β€˜(1st β€˜(𝑑 + 𝑓))), (𝑠 ∘ (2nd β€˜(𝑑 + 𝑓)))⟩ = ⟨((1st β€˜(𝑠 Β· 𝑑)) ∘ (1st β€˜(𝑠 Β· 𝑓))), ((2nd β€˜(𝑠 Β· 𝑑)) ⨣ (2nd β€˜(𝑠 Β· 𝑓)))⟩)
1111, 2, 3, 4, 10, 17, 8dvhvaddcl 40058 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑑 + 𝑓) ∈ (𝑇 Γ— 𝐸))
1121113adantr1 1169 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑑 + 𝑓) ∈ (𝑇 Γ— 𝐸))
1131, 2, 3, 4, 12dvhvsca 40064 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ (𝑑 + 𝑓) ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 Β· (𝑑 + 𝑓)) = ⟨(π‘ β€˜(1st β€˜(𝑑 + 𝑓))), (𝑠 ∘ (2nd β€˜(𝑑 + 𝑓)))⟩)
11437, 38, 112, 113syl12anc 835 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 Β· (𝑑 + 𝑓)) = ⟨(π‘ β€˜(1st β€˜(𝑑 + 𝑓))), (𝑠 ∘ (2nd β€˜(𝑑 + 𝑓)))⟩)
115353adantr3 1171 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 Β· 𝑑) ∈ (𝑇 Γ— 𝐸))
1161, 2, 3, 4, 12dvhvscacl 40066 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 Β· 𝑓) ∈ (𝑇 Γ— 𝐸))
1171163adantr2 1170 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 Β· 𝑓) ∈ (𝑇 Γ— 𝐸))
1181, 2, 3, 4, 10, 8, 17dvhvadd 40055 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑠 Β· 𝑑) ∈ (𝑇 Γ— 𝐸) ∧ (𝑠 Β· 𝑓) ∈ (𝑇 Γ— 𝐸))) β†’ ((𝑠 Β· 𝑑) + (𝑠 Β· 𝑓)) = ⟨((1st β€˜(𝑠 Β· 𝑑)) ∘ (1st β€˜(𝑠 Β· 𝑓))), ((2nd β€˜(𝑠 Β· 𝑑)) ⨣ (2nd β€˜(𝑠 Β· 𝑓)))⟩)
11937, 115, 117, 118syl12anc 835 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((𝑠 Β· 𝑑) + (𝑠 Β· 𝑓)) = ⟨((1st β€˜(𝑠 Β· 𝑑)) ∘ (1st β€˜(𝑠 Β· 𝑓))), ((2nd β€˜(𝑠 Β· 𝑑)) ⨣ (2nd β€˜(𝑠 Β· 𝑓)))⟩)
120110, 114, 1193eqtr4d 2782 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 Β· (𝑑 + 𝑓)) = ((𝑠 Β· 𝑑) + (𝑠 Β· 𝑓)))
121 simpl 483 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
122 simpr1 1194 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ 𝑠 ∈ 𝐸)
123 simpr2 1195 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ 𝑑 ∈ 𝐸)
124 simpr3 1196 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ 𝑓 ∈ (𝑇 Γ— 𝐸))
125124, 43syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (1st β€˜π‘“) ∈ 𝑇)
126 eqid 2732 . . . . . . . 8 (+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š)) = (+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š))
1271, 2, 3, 21, 126erngplus2 39767 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ (1st β€˜π‘“) ∈ 𝑇)) β†’ ((𝑠(+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š))𝑑)β€˜(1st β€˜π‘“)) = ((π‘ β€˜(1st β€˜π‘“)) ∘ (π‘‘β€˜(1st β€˜π‘“))))
128121, 122, 123, 125, 127syl13anc 1372 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((𝑠(+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š))𝑑)β€˜(1st β€˜π‘“)) = ((π‘ β€˜(1st β€˜π‘“)) ∘ (π‘‘β€˜(1st β€˜π‘“))))
12922fveq2d 6895 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (+gβ€˜π·) = (+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š)))
13017, 129eqtrid 2784 . . . . . . . . 9 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ⨣ = (+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š)))
131130oveqd 7428 . . . . . . . 8 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (𝑠 ⨣ 𝑑) = (𝑠(+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š))𝑑))
132131fveq1d 6893 . . . . . . 7 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ((𝑠 ⨣ 𝑑)β€˜(1st β€˜π‘“)) = ((𝑠(+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š))𝑑)β€˜(1st β€˜π‘“)))
133132adantr 481 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((𝑠 ⨣ 𝑑)β€˜(1st β€˜π‘“)) = ((𝑠(+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š))𝑑)β€˜(1st β€˜π‘“)))
134663adantr2 1170 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 Β· 𝑓) = ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)
135134fveq2d 6895 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (1st β€˜(𝑠 Β· 𝑓)) = (1st β€˜βŸ¨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩))
136135, 72eqtrdi 2788 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (1st β€˜(𝑠 Β· 𝑓)) = (π‘ β€˜(1st β€˜π‘“)))
1371, 2, 3, 4, 12dvhvsca 40064 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑑 Β· 𝑓) = ⟨(π‘‘β€˜(1st β€˜π‘“)), (𝑑 ∘ (2nd β€˜π‘“))⟩)
1381373adantr1 1169 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑑 Β· 𝑓) = ⟨(π‘‘β€˜(1st β€˜π‘“)), (𝑑 ∘ (2nd β€˜π‘“))⟩)
139138fveq2d 6895 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (1st β€˜(𝑑 Β· 𝑓)) = (1st β€˜βŸ¨(π‘‘β€˜(1st β€˜π‘“)), (𝑑 ∘ (2nd β€˜π‘“))⟩))
140 fvex 6904 . . . . . . . . 9 (π‘‘β€˜(1st β€˜π‘“)) ∈ V
141 vex 3478 . . . . . . . . . 10 𝑑 ∈ V
142141, 70coex 7923 . . . . . . . . 9 (𝑑 ∘ (2nd β€˜π‘“)) ∈ V
143140, 142op1st 7985 . . . . . . . 8 (1st β€˜βŸ¨(π‘‘β€˜(1st β€˜π‘“)), (𝑑 ∘ (2nd β€˜π‘“))⟩) = (π‘‘β€˜(1st β€˜π‘“))
144139, 143eqtrdi 2788 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (1st β€˜(𝑑 Β· 𝑓)) = (π‘‘β€˜(1st β€˜π‘“)))
145136, 144coeq12d 5864 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((1st β€˜(𝑠 Β· 𝑓)) ∘ (1st β€˜(𝑑 Β· 𝑓))) = ((π‘ β€˜(1st β€˜π‘“)) ∘ (π‘‘β€˜(1st β€˜π‘“))))
146128, 133, 1453eqtr4d 2782 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((𝑠 ⨣ 𝑑)β€˜(1st β€˜π‘“)) = ((1st β€˜(𝑠 Β· 𝑓)) ∘ (1st β€˜(𝑑 Β· 𝑓))))
14730adantr 481 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ 𝐷 ∈ Ring)
14816adantr 481 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ 𝐸 = (Baseβ€˜π·))
149122, 148eleqtrd 2835 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ 𝑠 ∈ (Baseβ€˜π·))
150123, 148eleqtrd 2835 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ 𝑑 ∈ (Baseβ€˜π·))
151124, 82syl 17 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (2nd β€˜π‘“) ∈ 𝐸)
152151, 148eleqtrd 2835 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (2nd β€˜π‘“) ∈ (Baseβ€˜π·))
15314, 17, 19ringdir 20084 . . . . . . . 8 ((𝐷 ∈ Ring ∧ (𝑠 ∈ (Baseβ€˜π·) ∧ 𝑑 ∈ (Baseβ€˜π·) ∧ (2nd β€˜π‘“) ∈ (Baseβ€˜π·))) β†’ ((𝑠 ⨣ 𝑑) Γ— (2nd β€˜π‘“)) = ((𝑠 Γ— (2nd β€˜π‘“)) ⨣ (𝑑 Γ— (2nd β€˜π‘“))))
154147, 149, 150, 152, 153syl13anc 1372 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((𝑠 ⨣ 𝑑) Γ— (2nd β€˜π‘“)) = ((𝑠 Γ— (2nd β€˜π‘“)) ⨣ (𝑑 Γ— (2nd β€˜π‘“))))
15514, 17ringacl 20097 . . . . . . . . . 10 ((𝐷 ∈ Ring ∧ 𝑠 ∈ (Baseβ€˜π·) ∧ 𝑑 ∈ (Baseβ€˜π·)) β†’ (𝑠 ⨣ 𝑑) ∈ (Baseβ€˜π·))
156147, 149, 150, 155syl3anc 1371 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 ⨣ 𝑑) ∈ (Baseβ€˜π·))
157156, 148eleqtrrd 2836 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 ⨣ 𝑑) ∈ 𝐸)
1581, 2, 3, 4, 10, 19dvhmulr 40049 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑠 ⨣ 𝑑) ∈ 𝐸 ∧ (2nd β€˜π‘“) ∈ 𝐸)) β†’ ((𝑠 ⨣ 𝑑) Γ— (2nd β€˜π‘“)) = ((𝑠 ⨣ 𝑑) ∘ (2nd β€˜π‘“)))
159121, 157, 151, 158syl12anc 835 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((𝑠 ⨣ 𝑑) Γ— (2nd β€˜π‘“)) = ((𝑠 ⨣ 𝑑) ∘ (2nd β€˜π‘“)))
160121, 122, 151, 94syl12anc 835 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 Γ— (2nd β€˜π‘“)) = (𝑠 ∘ (2nd β€˜π‘“)))
1611, 2, 3, 4, 10, 19dvhmulr 40049 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑑 ∈ 𝐸 ∧ (2nd β€˜π‘“) ∈ 𝐸)) β†’ (𝑑 Γ— (2nd β€˜π‘“)) = (𝑑 ∘ (2nd β€˜π‘“)))
162121, 123, 151, 161syl12anc 835 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑑 Γ— (2nd β€˜π‘“)) = (𝑑 ∘ (2nd β€˜π‘“)))
163160, 162oveq12d 7429 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((𝑠 Γ— (2nd β€˜π‘“)) ⨣ (𝑑 Γ— (2nd β€˜π‘“))) = ((𝑠 ∘ (2nd β€˜π‘“)) ⨣ (𝑑 ∘ (2nd β€˜π‘“))))
164154, 159, 1633eqtr3d 2780 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((𝑠 ⨣ 𝑑) ∘ (2nd β€˜π‘“)) = ((𝑠 ∘ (2nd β€˜π‘“)) ⨣ (𝑑 ∘ (2nd β€˜π‘“))))
165134fveq2d 6895 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (2nd β€˜(𝑠 Β· 𝑓)) = (2nd β€˜βŸ¨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩))
166165, 106eqtrdi 2788 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (2nd β€˜(𝑠 Β· 𝑓)) = (𝑠 ∘ (2nd β€˜π‘“)))
167138fveq2d 6895 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (2nd β€˜(𝑑 Β· 𝑓)) = (2nd β€˜βŸ¨(π‘‘β€˜(1st β€˜π‘“)), (𝑑 ∘ (2nd β€˜π‘“))⟩))
168140, 142op2nd 7986 . . . . . . . 8 (2nd β€˜βŸ¨(π‘‘β€˜(1st β€˜π‘“)), (𝑑 ∘ (2nd β€˜π‘“))⟩) = (𝑑 ∘ (2nd β€˜π‘“))
169167, 168eqtrdi 2788 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (2nd β€˜(𝑑 Β· 𝑓)) = (𝑑 ∘ (2nd β€˜π‘“)))
170166, 169oveq12d 7429 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((2nd β€˜(𝑠 Β· 𝑓)) ⨣ (2nd β€˜(𝑑 Β· 𝑓))) = ((𝑠 ∘ (2nd β€˜π‘“)) ⨣ (𝑑 ∘ (2nd β€˜π‘“))))
171164, 170eqtr4d 2775 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((𝑠 ⨣ 𝑑) ∘ (2nd β€˜π‘“)) = ((2nd β€˜(𝑠 Β· 𝑓)) ⨣ (2nd β€˜(𝑑 Β· 𝑓))))
172146, 171opeq12d 4881 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ⟨((𝑠 ⨣ 𝑑)β€˜(1st β€˜π‘“)), ((𝑠 ⨣ 𝑑) ∘ (2nd β€˜π‘“))⟩ = ⟨((1st β€˜(𝑠 Β· 𝑓)) ∘ (1st β€˜(𝑑 Β· 𝑓))), ((2nd β€˜(𝑠 Β· 𝑓)) ⨣ (2nd β€˜(𝑑 Β· 𝑓)))⟩)
1731, 2, 3, 4, 12dvhvsca 40064 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑠 ⨣ 𝑑) ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((𝑠 ⨣ 𝑑) Β· 𝑓) = ⟨((𝑠 ⨣ 𝑑)β€˜(1st β€˜π‘“)), ((𝑠 ⨣ 𝑑) ∘ (2nd β€˜π‘“))⟩)
174121, 157, 124, 173syl12anc 835 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((𝑠 ⨣ 𝑑) Β· 𝑓) = ⟨((𝑠 ⨣ 𝑑)β€˜(1st β€˜π‘“)), ((𝑠 ⨣ 𝑑) ∘ (2nd β€˜π‘“))⟩)
1751163adantr2 1170 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 Β· 𝑓) ∈ (𝑇 Γ— 𝐸))
1761, 2, 3, 4, 12dvhvscacl 40066 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑑 Β· 𝑓) ∈ (𝑇 Γ— 𝐸))
1771763adantr1 1169 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑑 Β· 𝑓) ∈ (𝑇 Γ— 𝐸))
1781, 2, 3, 4, 10, 8, 17dvhvadd 40055 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑠 Β· 𝑓) ∈ (𝑇 Γ— 𝐸) ∧ (𝑑 Β· 𝑓) ∈ (𝑇 Γ— 𝐸))) β†’ ((𝑠 Β· 𝑓) + (𝑑 Β· 𝑓)) = ⟨((1st β€˜(𝑠 Β· 𝑓)) ∘ (1st β€˜(𝑑 Β· 𝑓))), ((2nd β€˜(𝑠 Β· 𝑓)) ⨣ (2nd β€˜(𝑑 Β· 𝑓)))⟩)
179121, 175, 177, 178syl12anc 835 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((𝑠 Β· 𝑓) + (𝑑 Β· 𝑓)) = ⟨((1st β€˜(𝑠 Β· 𝑓)) ∘ (1st β€˜(𝑑 Β· 𝑓))), ((2nd β€˜(𝑠 Β· 𝑓)) ⨣ (2nd β€˜(𝑑 Β· 𝑓)))⟩)
180172, 174, 1793eqtr4d 2782 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((𝑠 ⨣ 𝑑) Β· 𝑓) = ((𝑠 Β· 𝑓) + (𝑑 Β· 𝑓)))
1811, 2, 3tendocoval 39729 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸) ∧ (1st β€˜π‘“) ∈ 𝑇) β†’ ((𝑠 ∘ 𝑑)β€˜(1st β€˜π‘“)) = (π‘ β€˜(π‘‘β€˜(1st β€˜π‘“))))
182121, 122, 123, 125, 181syl121anc 1375 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((𝑠 ∘ 𝑑)β€˜(1st β€˜π‘“)) = (π‘ β€˜(π‘‘β€˜(1st β€˜π‘“))))
183 coass 6264 . . . . . . 7 ((𝑠 ∘ 𝑑) ∘ (2nd β€˜π‘“)) = (𝑠 ∘ (𝑑 ∘ (2nd β€˜π‘“)))
184183a1i 11 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((𝑠 ∘ 𝑑) ∘ (2nd β€˜π‘“)) = (𝑠 ∘ (𝑑 ∘ (2nd β€˜π‘“))))
185182, 184opeq12d 4881 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ⟨((𝑠 ∘ 𝑑)β€˜(1st β€˜π‘“)), ((𝑠 ∘ 𝑑) ∘ (2nd β€˜π‘“))⟩ = ⟨(π‘ β€˜(π‘‘β€˜(1st β€˜π‘“))), (𝑠 ∘ (𝑑 ∘ (2nd β€˜π‘“)))⟩)
1861, 3tendococl 39735 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸) β†’ (𝑠 ∘ 𝑑) ∈ 𝐸)
187121, 122, 123, 186syl3anc 1371 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 ∘ 𝑑) ∈ 𝐸)
1881, 2, 3, 4, 12dvhvsca 40064 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑠 ∘ 𝑑) ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((𝑠 ∘ 𝑑) Β· 𝑓) = ⟨((𝑠 ∘ 𝑑)β€˜(1st β€˜π‘“)), ((𝑠 ∘ 𝑑) ∘ (2nd β€˜π‘“))⟩)
189121, 187, 124, 188syl12anc 835 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((𝑠 ∘ 𝑑) Β· 𝑓) = ⟨((𝑠 ∘ 𝑑)β€˜(1st β€˜π‘“)), ((𝑠 ∘ 𝑑) ∘ (2nd β€˜π‘“))⟩)
1901, 2, 3tendocl 39730 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑑 ∈ 𝐸 ∧ (1st β€˜π‘“) ∈ 𝑇) β†’ (π‘‘β€˜(1st β€˜π‘“)) ∈ 𝑇)
191121, 123, 125, 190syl3anc 1371 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (π‘‘β€˜(1st β€˜π‘“)) ∈ 𝑇)
1921, 3tendococl 39735 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑑 ∈ 𝐸 ∧ (2nd β€˜π‘“) ∈ 𝐸) β†’ (𝑑 ∘ (2nd β€˜π‘“)) ∈ 𝐸)
193121, 123, 151, 192syl3anc 1371 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑑 ∘ (2nd β€˜π‘“)) ∈ 𝐸)
1941, 2, 3, 4, 12dvhopvsca 40065 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ (π‘‘β€˜(1st β€˜π‘“)) ∈ 𝑇 ∧ (𝑑 ∘ (2nd β€˜π‘“)) ∈ 𝐸)) β†’ (𝑠 Β· ⟨(π‘‘β€˜(1st β€˜π‘“)), (𝑑 ∘ (2nd β€˜π‘“))⟩) = ⟨(π‘ β€˜(π‘‘β€˜(1st β€˜π‘“))), (𝑠 ∘ (𝑑 ∘ (2nd β€˜π‘“)))⟩)
195121, 122, 191, 193, 194syl13anc 1372 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 Β· ⟨(π‘‘β€˜(1st β€˜π‘“)), (𝑑 ∘ (2nd β€˜π‘“))⟩) = ⟨(π‘ β€˜(π‘‘β€˜(1st β€˜π‘“))), (𝑠 ∘ (𝑑 ∘ (2nd β€˜π‘“)))⟩)
196185, 189, 1953eqtr4d 2782 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((𝑠 ∘ 𝑑) Β· 𝑓) = (𝑠 Β· ⟨(π‘‘β€˜(1st β€˜π‘“)), (𝑑 ∘ (2nd β€˜π‘“))⟩))
1971, 2, 3, 4, 10, 19dvhmulr 40049 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸)) β†’ (𝑠 Γ— 𝑑) = (𝑠 ∘ 𝑑))
1981973adantr3 1171 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 Γ— 𝑑) = (𝑠 ∘ 𝑑))
199198oveq1d 7426 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((𝑠 Γ— 𝑑) Β· 𝑓) = ((𝑠 ∘ 𝑑) Β· 𝑓))
200138oveq2d 7427 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 Β· (𝑑 Β· 𝑓)) = (𝑠 Β· ⟨(π‘‘β€˜(1st β€˜π‘“)), (𝑑 ∘ (2nd β€˜π‘“))⟩))
201196, 199, 2003eqtr4d 2782 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ ((𝑠 Γ— 𝑑) Β· 𝑓) = (𝑠 Β· (𝑑 Β· 𝑓)))
202 xp1st 8009 . . . . . . 7 (𝑠 ∈ (𝑇 Γ— 𝐸) β†’ (1st β€˜π‘ ) ∈ 𝑇)
203202adantl 482 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑠 ∈ (𝑇 Γ— 𝐸)) β†’ (1st β€˜π‘ ) ∈ 𝑇)
204 fvresi 7173 . . . . . 6 ((1st β€˜π‘ ) ∈ 𝑇 β†’ (( I β†Ύ 𝑇)β€˜(1st β€˜π‘ )) = (1st β€˜π‘ ))
205203, 204syl 17 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑠 ∈ (𝑇 Γ— 𝐸)) β†’ (( I β†Ύ 𝑇)β€˜(1st β€˜π‘ )) = (1st β€˜π‘ ))
206 xp2nd 8010 . . . . . . 7 (𝑠 ∈ (𝑇 Γ— 𝐸) β†’ (2nd β€˜π‘ ) ∈ 𝐸)
2071, 2, 3tendof 39726 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (2nd β€˜π‘ ) ∈ 𝐸) β†’ (2nd β€˜π‘ ):π‘‡βŸΆπ‘‡)
208206, 207sylan2 593 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑠 ∈ (𝑇 Γ— 𝐸)) β†’ (2nd β€˜π‘ ):π‘‡βŸΆπ‘‡)
209 fcoi2 6766 . . . . . 6 ((2nd β€˜π‘ ):π‘‡βŸΆπ‘‡ β†’ (( I β†Ύ 𝑇) ∘ (2nd β€˜π‘ )) = (2nd β€˜π‘ ))
210208, 209syl 17 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑠 ∈ (𝑇 Γ— 𝐸)) β†’ (( I β†Ύ 𝑇) ∘ (2nd β€˜π‘ )) = (2nd β€˜π‘ ))
211205, 210opeq12d 4881 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑠 ∈ (𝑇 Γ— 𝐸)) β†’ ⟨(( I β†Ύ 𝑇)β€˜(1st β€˜π‘ )), (( I β†Ύ 𝑇) ∘ (2nd β€˜π‘ ))⟩ = ⟨(1st β€˜π‘ ), (2nd β€˜π‘ )⟩)
2121, 2, 3tendoidcl 39732 . . . . . 6 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ( I β†Ύ 𝑇) ∈ 𝐸)
213212anim1i 615 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑠 ∈ (𝑇 Γ— 𝐸)) β†’ (( I β†Ύ 𝑇) ∈ 𝐸 ∧ 𝑠 ∈ (𝑇 Γ— 𝐸)))
2141, 2, 3, 4, 12dvhvsca 40064 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (( I β†Ύ 𝑇) ∈ 𝐸 ∧ 𝑠 ∈ (𝑇 Γ— 𝐸))) β†’ (( I β†Ύ 𝑇) Β· 𝑠) = ⟨(( I β†Ύ 𝑇)β€˜(1st β€˜π‘ )), (( I β†Ύ 𝑇) ∘ (2nd β€˜π‘ ))⟩)
215213, 214syldan 591 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑠 ∈ (𝑇 Γ— 𝐸)) β†’ (( I β†Ύ 𝑇) Β· 𝑠) = ⟨(( I β†Ύ 𝑇)β€˜(1st β€˜π‘ )), (( I β†Ύ 𝑇) ∘ (2nd β€˜π‘ ))⟩)
216 1st2nd2 8016 . . . . 5 (𝑠 ∈ (𝑇 Γ— 𝐸) β†’ 𝑠 = ⟨(1st β€˜π‘ ), (2nd β€˜π‘ )⟩)
217216adantl 482 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑠 ∈ (𝑇 Γ— 𝐸)) β†’ 𝑠 = ⟨(1st β€˜π‘ ), (2nd β€˜π‘ )⟩)
218211, 215, 2173eqtr4d 2782 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑠 ∈ (𝑇 Γ— 𝐸)) β†’ (( I β†Ύ 𝑇) Β· 𝑠) = 𝑠)
2197, 9, 11, 13, 16, 18, 20, 26, 30, 34, 36, 120, 180, 201, 218islmodd 20481 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘ˆ ∈ LMod)
22010islvec 20720 . 2 (π‘ˆ ∈ LVec ↔ (π‘ˆ ∈ LMod ∧ 𝐷 ∈ DivRing))
221219, 28, 220sylanbrc 583 1 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘ˆ ∈ LVec)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βŸ¨cop 4634   I cid 5573   Γ— cxp 5674   β†Ύ cres 5678   ∘ ccom 5680  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411  1st c1st 7975  2nd c2nd 7976  Basecbs 17146  +gcplusg 17199  .rcmulr 17200  Scalarcsca 17202   ·𝑠 cvsca 17203  0gc0g 17387  invgcminusg 18822  1rcur 20006  Ringcrg 20058  DivRingcdr 20361  LModclmod 20475  LVecclvec 20718  HLchlt 38312  LHypclh 38947  LTrncltrn 39064  TEndoctendo 39715  EDRingcedring 39716  DVecHcdvh 40041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-riotaBAD 37915
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-tpos 8213  df-undef 8260  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-n0 12475  df-z 12561  df-uz 12825  df-fz 13487  df-struct 17082  df-sets 17099  df-slot 17117  df-ndx 17129  df-base 17147  df-ress 17176  df-plusg 17212  df-mulr 17213  df-sca 17215  df-vsca 17216  df-0g 17389  df-proset 18250  df-poset 18268  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-p1 18381  df-lat 18387  df-clat 18454  df-mgm 18563  df-sgrp 18612  df-mnd 18628  df-grp 18824  df-minusg 18825  df-mgp 19990  df-ur 20007  df-ring 20060  df-oppr 20154  df-dvdsr 20175  df-unit 20176  df-invr 20206  df-dvr 20219  df-drng 20363  df-lmod 20477  df-lvec 20719  df-oposet 38138  df-ol 38140  df-oml 38141  df-covers 38228  df-ats 38229  df-atl 38260  df-cvlat 38284  df-hlat 38313  df-llines 38461  df-lplanes 38462  df-lvols 38463  df-lines 38464  df-psubsp 38466  df-pmap 38467  df-padd 38759  df-lhyp 38951  df-laut 38952  df-ldil 39067  df-ltrn 39068  df-trl 39122  df-tendo 39718  df-edring 39720  df-dvech 40042
This theorem is referenced by:  dvhlvec  40072
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