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Mirrors > Home > MPE Home > Th. List > Mathboxes > tendoplcom | Structured version Visualization version GIF version |
Description: The endomorphism sum operation is commutative. (Contributed by NM, 11-Jun-2013.) |
Ref | Expression |
---|---|
tendopl.h | ⊢ 𝐻 = (LHyp‘𝐾) |
tendopl.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
tendopl.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
tendopl.p | ⊢ 𝑃 = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))) |
Ref | Expression |
---|---|
tendoplcom | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) → (𝑈𝑃𝑉) = (𝑉𝑃𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1135 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | tendopl.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | tendopl.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
4 | tendopl.e | . . 3 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
5 | tendopl.p | . . 3 ⊢ 𝑃 = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))) | |
6 | 2, 3, 4, 5 | tendoplcl 39968 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) → (𝑈𝑃𝑉) ∈ 𝐸) |
7 | 2, 3, 4, 5 | tendoplcl 39968 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑉 ∈ 𝐸 ∧ 𝑈 ∈ 𝐸) → (𝑉𝑃𝑈) ∈ 𝐸) |
8 | 7 | 3com23 1125 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) → (𝑉𝑃𝑈) ∈ 𝐸) |
9 | simpl1 1190 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑔 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | simpl2 1191 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑔 ∈ 𝑇) → 𝑈 ∈ 𝐸) | |
11 | simpr 484 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑔 ∈ 𝑇) → 𝑔 ∈ 𝑇) | |
12 | 2, 3, 4 | tendocl 39954 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇) → (𝑈‘𝑔) ∈ 𝑇) |
13 | 9, 10, 11, 12 | syl3anc 1370 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑔 ∈ 𝑇) → (𝑈‘𝑔) ∈ 𝑇) |
14 | simpl3 1192 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑔 ∈ 𝑇) → 𝑉 ∈ 𝐸) | |
15 | 2, 3, 4 | tendocl 39954 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑉 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇) → (𝑉‘𝑔) ∈ 𝑇) |
16 | 9, 14, 11, 15 | syl3anc 1370 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑔 ∈ 𝑇) → (𝑉‘𝑔) ∈ 𝑇) |
17 | 2, 3 | ltrncom 39925 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈‘𝑔) ∈ 𝑇 ∧ (𝑉‘𝑔) ∈ 𝑇) → ((𝑈‘𝑔) ∘ (𝑉‘𝑔)) = ((𝑉‘𝑔) ∘ (𝑈‘𝑔))) |
18 | 9, 13, 16, 17 | syl3anc 1370 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑔 ∈ 𝑇) → ((𝑈‘𝑔) ∘ (𝑉‘𝑔)) = ((𝑉‘𝑔) ∘ (𝑈‘𝑔))) |
19 | 5, 3 | tendopl2 39964 | . . . . 5 ⊢ ((𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇) → ((𝑈𝑃𝑉)‘𝑔) = ((𝑈‘𝑔) ∘ (𝑉‘𝑔))) |
20 | 10, 14, 11, 19 | syl3anc 1370 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑔 ∈ 𝑇) → ((𝑈𝑃𝑉)‘𝑔) = ((𝑈‘𝑔) ∘ (𝑉‘𝑔))) |
21 | 5, 3 | tendopl2 39964 | . . . . 5 ⊢ ((𝑉 ∈ 𝐸 ∧ 𝑈 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇) → ((𝑉𝑃𝑈)‘𝑔) = ((𝑉‘𝑔) ∘ (𝑈‘𝑔))) |
22 | 14, 10, 11, 21 | syl3anc 1370 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑔 ∈ 𝑇) → ((𝑉𝑃𝑈)‘𝑔) = ((𝑉‘𝑔) ∘ (𝑈‘𝑔))) |
23 | 18, 20, 22 | 3eqtr4d 2781 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑔 ∈ 𝑇) → ((𝑈𝑃𝑉)‘𝑔) = ((𝑉𝑃𝑈)‘𝑔)) |
24 | 23 | ralrimiva 3145 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) → ∀𝑔 ∈ 𝑇 ((𝑈𝑃𝑉)‘𝑔) = ((𝑉𝑃𝑈)‘𝑔)) |
25 | 2, 3, 4 | tendoeq1 39951 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑈𝑃𝑉) ∈ 𝐸 ∧ (𝑉𝑃𝑈) ∈ 𝐸) ∧ ∀𝑔 ∈ 𝑇 ((𝑈𝑃𝑉)‘𝑔) = ((𝑉𝑃𝑈)‘𝑔)) → (𝑈𝑃𝑉) = (𝑉𝑃𝑈)) |
26 | 1, 6, 8, 24, 25 | syl121anc 1374 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) → (𝑈𝑃𝑉) = (𝑉𝑃𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ↦ cmpt 5231 ∘ ccom 5680 ‘cfv 6543 (class class class)co 7412 ∈ cmpo 7414 HLchlt 38536 LHypclh 39171 LTrncltrn 39288 TEndoctendo 39939 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-riotaBAD 38139 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-undef 8264 df-map 8828 df-proset 18255 df-poset 18273 df-plt 18290 df-lub 18306 df-glb 18307 df-join 18308 df-meet 18309 df-p0 18385 df-p1 18386 df-lat 18392 df-clat 18459 df-oposet 38362 df-ol 38364 df-oml 38365 df-covers 38452 df-ats 38453 df-atl 38484 df-cvlat 38508 df-hlat 38537 df-llines 38685 df-lplanes 38686 df-lvols 38687 df-lines 38688 df-psubsp 38690 df-pmap 38691 df-padd 38983 df-lhyp 39175 df-laut 39176 df-ldil 39291 df-ltrn 39292 df-trl 39346 df-tendo 39942 |
This theorem is referenced by: tendo0plr 39979 tendoipl2 39985 erngdvlem2N 40176 erngdvlem2-rN 40184 dvhvaddcomN 40283 |
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