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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 1172 | . 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 36856 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) → (𝑈𝑃𝑉) ∈ 𝐸) |
7 | 2, 3, 4, 5 | tendoplcl 36856 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑉 ∈ 𝐸 ∧ 𝑈 ∈ 𝐸) → (𝑉𝑃𝑈) ∈ 𝐸) |
8 | 7 | 3com23 1162 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) → (𝑉𝑃𝑈) ∈ 𝐸) |
9 | simpl1 1248 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑔 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | simpl2 1250 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑔 ∈ 𝑇) → 𝑈 ∈ 𝐸) | |
11 | simpr 479 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑔 ∈ 𝑇) → 𝑔 ∈ 𝑇) | |
12 | 2, 3, 4 | tendocl 36842 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇) → (𝑈‘𝑔) ∈ 𝑇) |
13 | 9, 10, 11, 12 | syl3anc 1496 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑔 ∈ 𝑇) → (𝑈‘𝑔) ∈ 𝑇) |
14 | simpl3 1252 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑔 ∈ 𝑇) → 𝑉 ∈ 𝐸) | |
15 | 2, 3, 4 | tendocl 36842 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑉 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇) → (𝑉‘𝑔) ∈ 𝑇) |
16 | 9, 14, 11, 15 | syl3anc 1496 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑔 ∈ 𝑇) → (𝑉‘𝑔) ∈ 𝑇) |
17 | 2, 3 | ltrncom 36813 | . . . . 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 1496 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑔 ∈ 𝑇) → ((𝑈‘𝑔) ∘ (𝑉‘𝑔)) = ((𝑉‘𝑔) ∘ (𝑈‘𝑔))) |
19 | 5, 3 | tendopl2 36852 | . . . . 5 ⊢ ((𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇) → ((𝑈𝑃𝑉)‘𝑔) = ((𝑈‘𝑔) ∘ (𝑉‘𝑔))) |
20 | 10, 14, 11, 19 | syl3anc 1496 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑔 ∈ 𝑇) → ((𝑈𝑃𝑉)‘𝑔) = ((𝑈‘𝑔) ∘ (𝑉‘𝑔))) |
21 | 5, 3 | tendopl2 36852 | . . . . 5 ⊢ ((𝑉 ∈ 𝐸 ∧ 𝑈 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇) → ((𝑉𝑃𝑈)‘𝑔) = ((𝑉‘𝑔) ∘ (𝑈‘𝑔))) |
22 | 14, 10, 11, 21 | syl3anc 1496 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑔 ∈ 𝑇) → ((𝑉𝑃𝑈)‘𝑔) = ((𝑉‘𝑔) ∘ (𝑈‘𝑔))) |
23 | 18, 20, 22 | 3eqtr4d 2871 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝑔 ∈ 𝑇) → ((𝑈𝑃𝑉)‘𝑔) = ((𝑉𝑃𝑈)‘𝑔)) |
24 | 23 | ralrimiva 3175 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) → ∀𝑔 ∈ 𝑇 ((𝑈𝑃𝑉)‘𝑔) = ((𝑉𝑃𝑈)‘𝑔)) |
25 | 2, 3, 4 | tendoeq1 36839 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑈𝑃𝑉) ∈ 𝐸 ∧ (𝑉𝑃𝑈) ∈ 𝐸) ∧ ∀𝑔 ∈ 𝑇 ((𝑈𝑃𝑉)‘𝑔) = ((𝑉𝑃𝑈)‘𝑔)) → (𝑈𝑃𝑉) = (𝑉𝑃𝑈)) |
26 | 1, 6, 8, 24, 25 | syl121anc 1500 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) → (𝑈𝑃𝑉) = (𝑉𝑃𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ∀wral 3117 ↦ cmpt 4952 ∘ ccom 5346 ‘cfv 6123 (class class class)co 6905 ↦ cmpt2 6907 HLchlt 35425 LHypclh 36059 LTrncltrn 36176 TEndoctendo 36827 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-riotaBAD 35028 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-iin 4743 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-1st 7428 df-2nd 7429 df-undef 7664 df-map 8124 df-proset 17281 df-poset 17299 df-plt 17311 df-lub 17327 df-glb 17328 df-join 17329 df-meet 17330 df-p0 17392 df-p1 17393 df-lat 17399 df-clat 17461 df-oposet 35251 df-ol 35253 df-oml 35254 df-covers 35341 df-ats 35342 df-atl 35373 df-cvlat 35397 df-hlat 35426 df-llines 35573 df-lplanes 35574 df-lvols 35575 df-lines 35576 df-psubsp 35578 df-pmap 35579 df-padd 35871 df-lhyp 36063 df-laut 36064 df-ldil 36179 df-ltrn 36180 df-trl 36234 df-tendo 36830 |
This theorem is referenced by: tendo0plr 36867 tendoipl2 36873 erngdvlem2N 37064 erngdvlem2-rN 37072 dvhvaddcomN 37171 |
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