Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > tendo0plr | Structured version Visualization version GIF version |
Description: Property of the additive identity endormorphism. (Contributed by NM, 21-Feb-2014.) |
Ref | Expression |
---|---|
tendo0.b | ⊢ 𝐵 = (Base‘𝐾) |
tendo0.h | ⊢ 𝐻 = (LHyp‘𝐾) |
tendo0.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
tendo0.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
tendo0.o | ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
tendo0pl.p | ⊢ 𝑃 = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))) |
Ref | Expression |
---|---|
tendo0plr | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → (𝑆𝑃𝑂) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tendo0.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
2 | tendo0.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | tendo0.t | . . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
4 | tendo0.e | . . . . 5 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
5 | tendo0.o | . . . . 5 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
6 | 1, 2, 3, 4, 5 | tendo0cl 38498 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ 𝐸) |
7 | 6 | adantr 484 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → 𝑂 ∈ 𝐸) |
8 | tendo0pl.p | . . . 4 ⊢ 𝑃 = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))) | |
9 | 2, 3, 4, 8 | tendoplcom 38490 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑂 ∈ 𝐸) → (𝑆𝑃𝑂) = (𝑂𝑃𝑆)) |
10 | 7, 9 | mpd3an3 1464 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → (𝑆𝑃𝑂) = (𝑂𝑃𝑆)) |
11 | 1, 2, 3, 4, 5, 8 | tendo0pl 38499 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → (𝑂𝑃𝑆) = 𝑆) |
12 | 10, 11 | eqtrd 2774 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → (𝑆𝑃𝑂) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ↦ cmpt 5124 I cid 5443 ↾ cres 5542 ∘ ccom 5544 ‘cfv 6369 (class class class)co 7202 ∈ cmpo 7204 Basecbs 16684 HLchlt 37058 LHypclh 37692 LTrncltrn 37809 TEndoctendo 38460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-riotaBAD 36661 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-1st 7750 df-2nd 7751 df-undef 8004 df-map 8499 df-proset 17774 df-poset 17792 df-plt 17808 df-lub 17824 df-glb 17825 df-join 17826 df-meet 17827 df-p0 17903 df-p1 17904 df-lat 17910 df-clat 17977 df-oposet 36884 df-ol 36886 df-oml 36887 df-covers 36974 df-ats 36975 df-atl 37006 df-cvlat 37030 df-hlat 37059 df-llines 37206 df-lplanes 37207 df-lvols 37208 df-lines 37209 df-psubsp 37211 df-pmap 37212 df-padd 37504 df-lhyp 37696 df-laut 37697 df-ldil 37812 df-ltrn 37813 df-trl 37867 df-tendo 38463 |
This theorem is referenced by: cdlemn6 38910 dihopelvalcpre 38956 |
Copyright terms: Public domain | W3C validator |