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Theorem dvhlveclem 39502
Description: Lemma for dvhlvec 39503. 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 2738 . . . . 5 (Baseβ€˜π‘ˆ) = (Baseβ€˜π‘ˆ)
61, 2, 3, 4, 5dvhvbase 39481 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (Baseβ€˜π‘ˆ) = (𝑇 Γ— 𝐸))
76eqcomd 2744 . . 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 2738 . . . . 5 (Baseβ€˜π·) = (Baseβ€˜π·)
151, 3, 4, 10, 14dvhbase 39477 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (Baseβ€˜π·) = 𝐸)
1615eqcomd 2744 . . 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 2738 . . . . . 6 ((EDRingβ€˜πΎ)β€˜π‘Š) = ((EDRingβ€˜πΎ)β€˜π‘Š)
221, 21, 4, 10dvhsca 39476 . . . . 5 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 = ((EDRingβ€˜πΎ)β€˜π‘Š))
2322fveq2d 6842 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (1rβ€˜π·) = (1rβ€˜((EDRingβ€˜πΎ)β€˜π‘Š)))
24 eqid 2738 . . . . 5 (1rβ€˜((EDRingβ€˜πΎ)β€˜π‘Š)) = (1rβ€˜((EDRingβ€˜πΎ)β€˜π‘Š))
251, 2, 21, 24erng1r 39389 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (1rβ€˜((EDRingβ€˜πΎ)β€˜π‘Š)) = ( I β†Ύ 𝑇))
2623, 25eqtr2d 2779 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ( I β†Ύ 𝑇) = (1rβ€˜π·))
271, 21erngdv 39387 . . . . 5 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ((EDRingβ€˜πΎ)β€˜π‘Š) ∈ DivRing)
2822, 27eqeltrd 2839 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 ∈ DivRing)
29 drngring 20121 . . . 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 39501 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘ˆ ∈ Grp)
351, 2, 3, 4, 12dvhvscacl 39497 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 Β· 𝑑) ∈ (𝑇 Γ— 𝐸))
36353impb 1116 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸)) β†’ (𝑠 Β· 𝑑) ∈ (𝑇 Γ— 𝐸))
37 simpl 484 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
38 simpr1 1195 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ 𝑠 ∈ 𝐸)
39 simpr2 1196 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ 𝑑 ∈ (𝑇 Γ— 𝐸))
40 xp1st 7944 . . . . . . . 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 1197 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ 𝑓 ∈ (𝑇 Γ— 𝐸))
43 xp1st 7944 . . . . . . . 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 39414 . . . . . . 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 1373 . . . . . 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 39486 . . . . . . . . . 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 1170 . . . . . . . . 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 6842 . . . . . . . 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 6851 . . . . . . . . . 10 (1st β€˜π‘‘) ∈ V
51 fvex 6851 . . . . . . . . . 10 (1st β€˜π‘“) ∈ V
5250, 51coex 7858 . . . . . . . . 9 ((1st β€˜π‘‘) ∘ (1st β€˜π‘“)) ∈ V
53 ovex 7383 . . . . . . . . 9 ((2nd β€˜π‘‘) ⨣ (2nd β€˜π‘“)) ∈ V
5452, 53op1st 7920 . . . . . . . 8 (1st β€˜βŸ¨((1st β€˜π‘‘) ∘ (1st β€˜π‘“)), ((2nd β€˜π‘‘) ⨣ (2nd β€˜π‘“))⟩) = ((1st β€˜π‘‘) ∘ (1st β€˜π‘“))
5549, 54eqtrdi 2794 . . . . . . 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 6842 . . . . . 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 39495 . . . . . . . . . 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 1172 . . . . . . . . 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 6842 . . . . . . . 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 6851 . . . . . . . . 9 (π‘ β€˜(1st β€˜π‘‘)) ∈ V
61 vex 3448 . . . . . . . . . 10 𝑠 ∈ V
62 fvex 6851 . . . . . . . . . 10 (2nd β€˜π‘‘) ∈ V
6361, 62coex 7858 . . . . . . . . 9 (𝑠 ∘ (2nd β€˜π‘‘)) ∈ V
6460, 63op1st 7920 . . . . . . . 8 (1st β€˜βŸ¨(π‘ β€˜(1st β€˜π‘‘)), (𝑠 ∘ (2nd β€˜π‘‘))⟩) = (π‘ β€˜(1st β€˜π‘‘))
6559, 64eqtrdi 2794 . . . . . . 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 39495 . . . . . . . . . 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 1171 . . . . . . . . 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 6842 . . . . . . . 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 6851 . . . . . . . . 9 (π‘ β€˜(1st β€˜π‘“)) ∈ V
70 fvex 6851 . . . . . . . . . 10 (2nd β€˜π‘“) ∈ V
7161, 70coex 7858 . . . . . . . . 9 (𝑠 ∘ (2nd β€˜π‘“)) ∈ V
7269, 71op1st 7920 . . . . . . . 8 (1st β€˜βŸ¨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩) = (π‘ β€˜(1st β€˜π‘“))
7368, 72eqtrdi 2794 . . . . . . 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 5817 . . . . . 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 2788 . . . . 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 482 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ 𝐷 ∈ Ring)
7716adantr 482 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ 𝐸 = (Baseβ€˜π·))
7838, 77eleqtrd 2841 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ 𝑠 ∈ (Baseβ€˜π·))
79 xp2nd 7945 . . . . . . . . . 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 2841 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (2nd β€˜π‘‘) ∈ (Baseβ€˜π·))
82 xp2nd 7945 . . . . . . . . . 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 2841 . . . . . . . 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 19917 . . . . . . . 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 1373 . . . . . . 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 19929 . . . . . . . . . 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 1372 . . . . . . . . 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 2842 . . . . . . . 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 39480 . . . . . . . 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 836 . . . . . . 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 39480 . . . . . . . . 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 836 . . . . . . . 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 39480 . . . . . . . . 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 836 . . . . . . . 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 7368 . . . . . . 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 2786 . . . . . 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 6842 . . . . . . . 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 7921 . . . . . . . 8 (2nd β€˜βŸ¨((1st β€˜π‘‘) ∘ (1st β€˜π‘“)), ((2nd β€˜π‘‘) ⨣ (2nd β€˜π‘“))⟩) = ((2nd β€˜π‘‘) ⨣ (2nd β€˜π‘“))
10098, 99eqtrdi 2794 . . . . . . 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 5815 . . . . . 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 6842 . . . . . . . 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 7921 . . . . . . . 8 (2nd β€˜βŸ¨(π‘ β€˜(1st β€˜π‘‘)), (𝑠 ∘ (2nd β€˜π‘‘))⟩) = (𝑠 ∘ (2nd β€˜π‘‘))
104102, 103eqtrdi 2794 . . . . . . 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 6842 . . . . . . . 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 7921 . . . . . . . 8 (2nd β€˜βŸ¨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩) = (𝑠 ∘ (2nd β€˜π‘“))
107105, 106eqtrdi 2794 . . . . . . 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 7368 . . . . . 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 2788 . . . . 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 4837 . . . 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 39489 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑑 ∈ (𝑇 Γ— 𝐸) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑑 + 𝑓) ∈ (𝑇 Γ— 𝐸))
1121113adantr1 1170 . . . . 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 39495 . . . . 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 836 . . . 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 1172 . . . . 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 39497 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 Β· 𝑓) ∈ (𝑇 Γ— 𝐸))
1171163adantr2 1171 . . . . 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 39486 . . . . 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 836 . . . 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 2788 . . 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 484 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
122 simpr1 1195 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ 𝑠 ∈ 𝐸)
123 simpr2 1196 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ 𝑑 ∈ 𝐸)
124 simpr3 1197 . . . . . . . 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 2738 . . . . . . . 8 (+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š)) = (+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š))
1271, 2, 3, 21, 126erngplus2 39198 . . . . . . 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 1373 . . . . . 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 6842 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (+gβ€˜π·) = (+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š)))
13017, 129eqtrid 2790 . . . . . . . . 9 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ⨣ = (+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š)))
131130oveqd 7367 . . . . . . . 8 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (𝑠 ⨣ 𝑑) = (𝑠(+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š))𝑑))
132131fveq1d 6840 . . . . . . 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 482 . . . . . 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 1171 . . . . . . . . 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 6842 . . . . . . . 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 2794 . . . . . . 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 39495 . . . . . . . . . 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 1170 . . . . . . . . 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 6842 . . . . . . . 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 6851 . . . . . . . . 9 (π‘‘β€˜(1st β€˜π‘“)) ∈ V
141 vex 3448 . . . . . . . . . 10 𝑑 ∈ V
142141, 70coex 7858 . . . . . . . . 9 (𝑑 ∘ (2nd β€˜π‘“)) ∈ V
143140, 142op1st 7920 . . . . . . . 8 (1st β€˜βŸ¨(π‘‘β€˜(1st β€˜π‘“)), (𝑑 ∘ (2nd β€˜π‘“))⟩) = (π‘‘β€˜(1st β€˜π‘“))
144139, 143eqtrdi 2794 . . . . . . 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 5817 . . . . . 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 2788 . . . . 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 482 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ 𝐷 ∈ Ring)
14816adantr 482 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ 𝐸 = (Baseβ€˜π·))
149122, 148eleqtrd 2841 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ 𝑠 ∈ (Baseβ€˜π·))
150123, 148eleqtrd 2841 . . . . . . . 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 2841 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (2nd β€˜π‘“) ∈ (Baseβ€˜π·))
15314, 17, 19ringdir 19918 . . . . . . . 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 1373 . . . . . . 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 19929 . . . . . . . . . 10 ((𝐷 ∈ Ring ∧ 𝑠 ∈ (Baseβ€˜π·) ∧ 𝑑 ∈ (Baseβ€˜π·)) β†’ (𝑠 ⨣ 𝑑) ∈ (Baseβ€˜π·))
156147, 149, 150, 155syl3anc 1372 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 ⨣ 𝑑) ∈ (Baseβ€˜π·))
157156, 148eleqtrrd 2842 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 ⨣ 𝑑) ∈ 𝐸)
1581, 2, 3, 4, 10, 19dvhmulr 39480 . . . . . . . 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 836 . . . . . . 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 836 . . . . . . . 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 39480 . . . . . . . . 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 836 . . . . . . . 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 7368 . . . . . . 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 2786 . . . . . 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 6842 . . . . . . . 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 2794 . . . . . . 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 6842 . . . . . . . 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 7921 . . . . . . . 8 (2nd β€˜βŸ¨(π‘‘β€˜(1st β€˜π‘“)), (𝑑 ∘ (2nd β€˜π‘“))⟩) = (𝑑 ∘ (2nd β€˜π‘“))
169167, 168eqtrdi 2794 . . . . . . 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 7368 . . . . . 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 2781 . . . . 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 4837 . . . 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 39495 . . . . 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 836 . . . 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 1171 . . . . 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 39497 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑑 Β· 𝑓) ∈ (𝑇 Γ— 𝐸))
1771763adantr1 1170 . . . . 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 39486 . . . . 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 836 . . . 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 2788 . . 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 39160 . . . . . . 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 1376 . . . . . 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 6214 . . . . . . 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 4837 . . . . 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 39166 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸) β†’ (𝑠 ∘ 𝑑) ∈ 𝐸)
187121, 122, 123, 186syl3anc 1372 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 ∘ 𝑑) ∈ 𝐸)
1881, 2, 3, 4, 12dvhvsca 39495 . . . . . 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 836 . . . . 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 39161 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑑 ∈ 𝐸 ∧ (1st β€˜π‘“) ∈ 𝑇) β†’ (π‘‘β€˜(1st β€˜π‘“)) ∈ 𝑇)
191121, 123, 125, 190syl3anc 1372 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (π‘‘β€˜(1st β€˜π‘“)) ∈ 𝑇)
1921, 3tendococl 39166 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑑 ∈ 𝐸 ∧ (2nd β€˜π‘“) ∈ 𝐸) β†’ (𝑑 ∘ (2nd β€˜π‘“)) ∈ 𝐸)
193121, 123, 151, 192syl3anc 1372 . . . . . 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 39496 . . . . . 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 1373 . . . . 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 2788 . . . 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 39480 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸)) β†’ (𝑠 Γ— 𝑑) = (𝑠 ∘ 𝑑))
1981973adantr3 1172 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸 ∧ 𝑓 ∈ (𝑇 Γ— 𝐸))) β†’ (𝑠 Γ— 𝑑) = (𝑠 ∘ 𝑑))
199198oveq1d 7365 . . . 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 7366 . . . 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 2788 . . 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 7944 . . . . . . 7 (𝑠 ∈ (𝑇 Γ— 𝐸) β†’ (1st β€˜π‘ ) ∈ 𝑇)
203202adantl 483 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑠 ∈ (𝑇 Γ— 𝐸)) β†’ (1st β€˜π‘ ) ∈ 𝑇)
204 fvresi 7114 . . . . . 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 7945 . . . . . . 7 (𝑠 ∈ (𝑇 Γ— 𝐸) β†’ (2nd β€˜π‘ ) ∈ 𝐸)
2071, 2, 3tendof 39157 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (2nd β€˜π‘ ) ∈ 𝐸) β†’ (2nd β€˜π‘ ):π‘‡βŸΆπ‘‡)
208206, 207sylan2 594 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑠 ∈ (𝑇 Γ— 𝐸)) β†’ (2nd β€˜π‘ ):π‘‡βŸΆπ‘‡)
209 fcoi2 6713 . . . . . 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 4837 . . . 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 39163 . . . . . 6 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ( I β†Ύ 𝑇) ∈ 𝐸)
213212anim1i 616 . . . . 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 39495 . . . . 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 592 . . . 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 7951 . . . . 5 (𝑠 ∈ (𝑇 Γ— 𝐸) β†’ 𝑠 = ⟨(1st β€˜π‘ ), (2nd β€˜π‘ )⟩)
217216adantl 483 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑠 ∈ (𝑇 Γ— 𝐸)) β†’ 𝑠 = ⟨(1st β€˜π‘ ), (2nd β€˜π‘ )⟩)
218211, 215, 2173eqtr4d 2788 . . 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 20257 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘ˆ ∈ LMod)
22010islvec 20494 . 2 (π‘ˆ ∈ LVec ↔ (π‘ˆ ∈ LMod ∧ 𝐷 ∈ DivRing))
221219, 28, 220sylanbrc 584 1 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘ˆ ∈ LVec)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βŸ¨cop 4591   I cid 5528   Γ— cxp 5629   β†Ύ cres 5633   ∘ ccom 5635  βŸΆwf 6488  β€˜cfv 6492  (class class class)co 7350  1st c1st 7910  2nd c2nd 7911  Basecbs 17019  +gcplusg 17069  .rcmulr 17070  Scalarcsca 17072   ·𝑠 cvsca 17073  0gc0g 17257  invgcminusg 18685  1rcur 19848  Ringcrg 19894  DivRingcdr 20114  LModclmod 20251  LVecclvec 20492  HLchlt 37743  LHypclh 38378  LTrncltrn 38495  TEndoctendo 39146  EDRingcedring 39147  DVecHcdvh 39472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7663  ax-cnex 11041  ax-resscn 11042  ax-1cn 11043  ax-icn 11044  ax-addcl 11045  ax-addrcl 11046  ax-mulcl 11047  ax-mulrcl 11048  ax-mulcom 11049  ax-addass 11050  ax-mulass 11051  ax-distr 11052  ax-i2m1 11053  ax-1ne0 11054  ax-1rid 11055  ax-rnegex 11056  ax-rrecex 11057  ax-cnre 11058  ax-pre-lttri 11059  ax-pre-lttrn 11060  ax-pre-ltadd 11061  ax-pre-mulgt0 11062  ax-riotaBAD 37346
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4865  df-iun 4955  df-iin 4956  df-br 5105  df-opab 5167  df-mpt 5188  df-tr 5222  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6250  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7306  df-ov 7353  df-oprab 7354  df-mpo 7355  df-om 7794  df-1st 7912  df-2nd 7913  df-tpos 8125  df-undef 8172  df-frecs 8180  df-wrecs 8211  df-recs 8285  df-rdg 8324  df-1o 8380  df-er 8582  df-map 8701  df-en 8818  df-dom 8819  df-sdom 8820  df-fin 8821  df-pnf 11125  df-mnf 11126  df-xr 11127  df-ltxr 11128  df-le 11129  df-sub 11321  df-neg 11322  df-nn 12088  df-2 12150  df-3 12151  df-4 12152  df-5 12153  df-6 12154  df-n0 12348  df-z 12434  df-uz 12698  df-fz 13355  df-struct 16955  df-sets 16972  df-slot 16990  df-ndx 17002  df-base 17020  df-ress 17049  df-plusg 17082  df-mulr 17083  df-sca 17085  df-vsca 17086  df-0g 17259  df-proset 18120  df-poset 18138  df-plt 18155  df-lub 18171  df-glb 18172  df-join 18173  df-meet 18174  df-p0 18250  df-p1 18251  df-lat 18257  df-clat 18324  df-mgm 18433  df-sgrp 18482  df-mnd 18493  df-grp 18687  df-minusg 18688  df-mgp 19832  df-ur 19849  df-ring 19896  df-oppr 19978  df-dvdsr 19999  df-unit 20000  df-invr 20030  df-dvr 20041  df-drng 20116  df-lmod 20253  df-lvec 20493  df-oposet 37569  df-ol 37571  df-oml 37572  df-covers 37659  df-ats 37660  df-atl 37691  df-cvlat 37715  df-hlat 37744  df-llines 37892  df-lplanes 37893  df-lvols 37894  df-lines 37895  df-psubsp 37897  df-pmap 37898  df-padd 38190  df-lhyp 38382  df-laut 38383  df-ldil 38498  df-ltrn 38499  df-trl 38553  df-tendo 39149  df-edring 39151  df-dvech 39473
This theorem is referenced by:  dvhlvec  39503
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