dvhsca - Mathbox for Norm Megill

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Theorem dvhsca 38220

Description: The ring of scalars of the constructed full vector space H. (Contributed by NM, 22-Jun-2014.)

**Hypotheses**
Ref	Expression
dvhsca.h	⊢ 𝐻 = (LHyp‘𝐾)
dvhsca.d	⊢ 𝐷 = ((EDRing‘𝐾)‘𝑊)
dvhsca.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dvhsca.f	⊢ 𝐹 = (Scalar‘𝑈)

**Assertion**
Ref	Expression
dvhsca	⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝐹 = 𝐷)

Proof of Theorem dvhsca Dummy variables 𝑓 𝑔 ℎ 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvhsca.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		eqid 2823	. . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
3		eqid 2823	. . . 4 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
4		dvhsca.d	. . . 4 ⊢ 𝐷 = ((EDRing‘𝐾)‘𝑊)
5		dvhsca.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	dvhset 38219	. . 3 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝑈 = ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊))⟩, ⟨(+_g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ (((2^nd ‘𝑓)‘ℎ) ∘ ((2^nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·_𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑓 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)⟩}))
7	6	fveq2d 6676	. 2 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → (Scalar‘𝑈) = (Scalar‘({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊))⟩, ⟨(+_g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ (((2^nd ‘𝑓)‘ℎ) ∘ ((2^nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·_𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑓 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)⟩})))
8		dvhsca.f	. 2 ⊢ 𝐹 = (Scalar‘𝑈)
9	4	fvexi 6686	. . 3 ⊢ 𝐷 ∈ V
10		eqid 2823	. . . 4 ⊢ ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊))⟩, ⟨(+_g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ (((2^nd ‘𝑓)‘ℎ) ∘ ((2^nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·_𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑓 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)⟩}) = ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊))⟩, ⟨(+_g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ (((2^nd ‘𝑓)‘ℎ) ∘ ((2^nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·_𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑓 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)⟩})
11	10	lmodsca 16641	. . 3 ⊢ (𝐷 ∈ V → 𝐷 = (Scalar‘({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊))⟩, ⟨(+_g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ (((2^nd ‘𝑓)‘ℎ) ∘ ((2^nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·_𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑓 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)⟩})))
12	9, 11	ax-mp 5	. 2 ⊢ 𝐷 = (Scalar‘({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊))⟩, ⟨(+_g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ (((2^nd ‘𝑓)‘ℎ) ∘ ((2^nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·_𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑓 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)⟩}))
13	7, 8, 12	3eqtr4g 2883	1 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝐹 = 𝐷)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∪ cun 3936 {csn 4569 {ctp 4573 ⟨cop 4575 ↦ cmpt 5148 × cxp 5555 ∘ ccom 5561 ‘cfv 6357 ∈ cmpo 7160 1^st c1st 7689 2^nd c2nd 7690 ndxcnx 16482 Basecbs 16485 +_gcplusg 16567 Scalarcsca 16570 ·_𝑠 cvsca 16571 LHypclh 37122 LTrncltrn 37239 TEndoctendo 37890 EDRingcedring 37891 DVecHcdvh 38216

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-plusg 16580 df-sca 16583 df-vsca 16584 df-dvech 38217

This theorem is referenced by: dvhbase 38221 dvhfplusr 38222 dvhfmulr 38223 dvhfvadd 38229 dvhvaddass 38235 tendoinvcl 38242 tendolinv 38243 tendorinv 38244 dvhgrp 38245 dvhlveclem 38246 cdlemn4 38336 hlhilsbase2 39080 hlhilsplus2 39081 hlhilsmul2 39082

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