tendoipl2 - Mathbox for Norm Megill

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Theorem tendoipl2 40785

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 40783	. . 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 40769	. . 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 40784	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → ((𝐼‘𝑆)𝑃𝑆) = 𝑂)
12	8, 11	eqtrd 2764	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 5183 I cid 5525 ^◡ccnv 5630 ↾ cres 5633 ∘ ccom 5635 ‘cfv 6499 (class class class)co 7369 ∈ cmpo 7371 Basecbs 17155 HLchlt 39336 LHypclh 39971 LTrncltrn 40088 TEndoctendo 40739

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 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-riotaBAD 38939

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5526 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-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-1st 7947 df-2nd 7948 df-undef 8229 df-map 8778 df-proset 18235 df-poset 18254 df-plt 18269 df-lub 18285 df-glb 18286 df-join 18287 df-meet 18288 df-p0 18364 df-p1 18365 df-lat 18373 df-clat 18440 df-oposet 39162 df-ol 39164 df-oml 39165 df-covers 39252 df-ats 39253 df-atl 39284 df-cvlat 39308 df-hlat 39337 df-llines 39485 df-lplanes 39486 df-lvols 39487 df-lines 39488 df-psubsp 39490 df-pmap 39491 df-padd 39783 df-lhyp 39975 df-laut 39976 df-ldil 40091 df-ltrn 40092 df-trl 40146 df-tendo 40742

This theorem is referenced by: dihjatcclem4 41408

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