tendoipl2 - Mathbox for Norm Megill

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Theorem tendoipl2 40787

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 40785	. . 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 40771	. . 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 40786	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → ((𝐼‘𝑆)𝑃𝑆) = 𝑂)
12	8, 11	eqtrd 2765	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → (𝑆𝑃(𝐼‘𝑆)) = 𝑂)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ↦ cmpt 5190 I cid 5534 ^◡ccnv 5639 ↾ cres 5642 ∘ ccom 5644 ‘cfv 6513 (class class class)co 7389 ∈ cmpo 7391 Basecbs 17185 HLchlt 39338 LHypclh 39973 LTrncltrn 40090 TEndoctendo 40741

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-riotaBAD 38941

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-iin 4960 df-br 5110 df-opab 5172 df-mpt 5191 df-id 5535 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-1st 7970 df-2nd 7971 df-undef 8254 df-map 8803 df-proset 18261 df-poset 18280 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-p1 18391 df-lat 18397 df-clat 18464 df-oposet 39164 df-ol 39166 df-oml 39167 df-covers 39254 df-ats 39255 df-atl 39286 df-cvlat 39310 df-hlat 39339 df-llines 39487 df-lplanes 39488 df-lvols 39489 df-lines 39490 df-psubsp 39492 df-pmap 39493 df-padd 39785 df-lhyp 39977 df-laut 39978 df-ldil 40093 df-ltrn 40094 df-trl 40148 df-tendo 40744

This theorem is referenced by: dihjatcclem4 41410

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