tendoipl2 - Mathbox for Norm Megill

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Theorem tendoipl2 40907

Description: Property of the additive inverse endomorphism. (Contributed by NM, 29-Sep-2014.)

**Hypotheses**
Ref	Expression
tendoicl.h	⊢ 𝐻 = (LHyp‘𝐾)
tendoicl.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
tendoicl.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
tendoicl.i	⊢ 𝐼 = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ^◡(𝑠‘𝑓)))
tendoi.b	⊢ 𝐵 = (Base‘𝐾)
tendoi.p	⊢ 𝑃 = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))
tendoi.o	⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))

**Assertion**
Ref	Expression
tendoipl2	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → (𝑆𝑃(𝐼‘𝑆)) = 𝑂)

Distinct variable groups: 𝐸,𝑠 𝑓,𝑠,𝑇 𝑓,𝑊,𝑠 𝐵,𝑓 𝑡,𝐸 𝑓,𝐻 𝑓,𝐾 𝑡,𝑓,𝑠,𝑇 𝑡,𝑊 Allowed substitution hints: 𝐵(𝑡,𝑠) 𝑃(𝑡,𝑓,𝑠) 𝑆(𝑡,𝑓,𝑠) 𝐸(𝑓) 𝐻(𝑡,𝑠) 𝐼(𝑡,𝑓,𝑠) 𝐾(𝑡,𝑠) 𝑂(𝑡,𝑓,𝑠)

**Proof of Theorem tendoipl2**
Step	Hyp	Ref	Expression
1		tendoicl.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		tendoicl.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
3		tendoicl.e	. . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
4		tendoicl.i	. . . 4 ⊢ 𝐼 = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ^◡(𝑠‘𝑓)))
5	1, 2, 3, 4	tendoicl 40905	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → (𝐼‘𝑆) ∈ 𝐸)
6		tendoi.p	. . . 4 ⊢ 𝑃 = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))
7	1, 2, 3, 6	tendoplcom 40891	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ (𝐼‘𝑆) ∈ 𝐸) → (𝑆𝑃(𝐼‘𝑆)) = ((𝐼‘𝑆)𝑃𝑆))
8	5, 7	mpd3an3 1464	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → (𝑆𝑃(𝐼‘𝑆)) = ((𝐼‘𝑆)𝑃𝑆))
9		tendoi.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
10		tendoi.o	. . 3 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))
11	1, 2, 3, 4, 9, 6, 10	tendoipl 40906	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → ((𝐼‘𝑆)𝑃𝑆) = 𝑂)
12	8, 11	eqtrd 2766	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → (𝑆𝑃(𝐼‘𝑆)) = 𝑂)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ↦ cmpt 5170 I cid 5508 ^◡ccnv 5613 ↾ cres 5616 ∘ ccom 5618 ‘cfv 6481 (class class class)co 7346 ∈ cmpo 7348 Basecbs 17120 HLchlt 39459 LHypclh 40093 LTrncltrn 40210 TEndoctendo 40861

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-riotaBAD 39062

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-1st 7921 df-2nd 7922 df-undef 8203 df-map 8752 df-proset 18200 df-poset 18219 df-plt 18234 df-lub 18250 df-glb 18251 df-join 18252 df-meet 18253 df-p0 18329 df-p1 18330 df-lat 18338 df-clat 18405 df-oposet 39285 df-ol 39287 df-oml 39288 df-covers 39375 df-ats 39376 df-atl 39407 df-cvlat 39431 df-hlat 39460 df-llines 39607 df-lplanes 39608 df-lvols 39609 df-lines 39610 df-psubsp 39612 df-pmap 39613 df-padd 39905 df-lhyp 40097 df-laut 40098 df-ldil 40213 df-ltrn 40214 df-trl 40268 df-tendo 40864

This theorem is referenced by: dihjatcclem4 41530

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