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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihopellsm | Structured version Visualization version GIF version |
Description: Ordered pair membership in a subspace sum of isomorphism H values. (Contributed by NM, 26-Sep-2014.) |
Ref | Expression |
---|---|
dihopellsm.b | ⊢ 𝐵 = (Base‘𝐾) |
dihopellsm.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihopellsm.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dihopellsm.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
dihopellsm.a | ⊢ 𝐴 = (𝑣 ∈ 𝐸, 𝑤 ∈ 𝐸 ↦ (𝑖 ∈ 𝑇 ↦ ((𝑣‘𝑖) ∘ (𝑤‘𝑖)))) |
dihopellsm.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dihopellsm.l | ⊢ 𝐿 = (LSubSp‘𝑈) |
dihopellsm.p | ⊢ ⊕ = (LSSum‘𝑈) |
dihopellsm.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dihopellsm.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dihopellsm.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
dihopellsm.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
dihopellsm | ⊢ (𝜑 → (〈𝐹, 𝑆〉 ∈ ((𝐼‘𝑋) ⊕ (𝐼‘𝑌)) ↔ ∃𝑔∃𝑡∃ℎ∃𝑢((〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋) ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) ∧ (𝐹 = (𝑔 ∘ ℎ) ∧ 𝑆 = (𝑡𝐴𝑢))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dihopellsm.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | dihopellsm.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | dihopellsm.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
4 | dihopellsm.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | dihopellsm.i | . . . . 5 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
6 | dihopellsm.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
7 | eqid 2772 | . . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
8 | 3, 4, 5, 6, 7 | dihlss 37831 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ (LSubSp‘𝑈)) |
9 | 1, 2, 8 | syl2anc 576 | . . 3 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (LSubSp‘𝑈)) |
10 | dihopellsm.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
11 | 3, 4, 5, 6, 7 | dihlss 37831 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵) → (𝐼‘𝑌) ∈ (LSubSp‘𝑈)) |
12 | 1, 10, 11 | syl2anc 576 | . . 3 ⊢ (𝜑 → (𝐼‘𝑌) ∈ (LSubSp‘𝑈)) |
13 | eqid 2772 | . . . 4 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
14 | dihopellsm.p | . . . 4 ⊢ ⊕ = (LSSum‘𝑈) | |
15 | 4, 6, 13, 7, 14 | dvhopellsm 37698 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐼‘𝑋) ∈ (LSubSp‘𝑈) ∧ (𝐼‘𝑌) ∈ (LSubSp‘𝑈)) → (〈𝐹, 𝑆〉 ∈ ((𝐼‘𝑋) ⊕ (𝐼‘𝑌)) ↔ ∃𝑔∃𝑡∃ℎ∃𝑢((〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋) ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) ∧ 〈𝐹, 𝑆〉 = (〈𝑔, 𝑡〉(+g‘𝑈)〈ℎ, 𝑢〉)))) |
16 | 1, 9, 12, 15 | syl3anc 1351 | . 2 ⊢ (𝜑 → (〈𝐹, 𝑆〉 ∈ ((𝐼‘𝑋) ⊕ (𝐼‘𝑌)) ↔ ∃𝑔∃𝑡∃ℎ∃𝑢((〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋) ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) ∧ 〈𝐹, 𝑆〉 = (〈𝑔, 𝑡〉(+g‘𝑈)〈ℎ, 𝑢〉)))) |
17 | dihopellsm.t | . . . . . . 7 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
18 | dihopellsm.e | . . . . . . 7 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
19 | 1 | adantr 473 | . . . . . . 7 ⊢ ((𝜑 ∧ 〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
20 | 2 | adantr 473 | . . . . . . 7 ⊢ ((𝜑 ∧ 〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋)) → 𝑋 ∈ 𝐵) |
21 | simpr 477 | . . . . . . 7 ⊢ ((𝜑 ∧ 〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋)) → 〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋)) | |
22 | 3, 4, 17, 18, 5, 19, 20, 21 | dihopcl 37834 | . . . . . 6 ⊢ ((𝜑 ∧ 〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋)) → (𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸)) |
23 | 1 | adantr 473 | . . . . . . 7 ⊢ ((𝜑 ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
24 | 10 | adantr 473 | . . . . . . 7 ⊢ ((𝜑 ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) → 𝑌 ∈ 𝐵) |
25 | simpr 477 | . . . . . . 7 ⊢ ((𝜑 ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) → 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) | |
26 | 3, 4, 17, 18, 5, 23, 24, 25 | dihopcl 37834 | . . . . . 6 ⊢ ((𝜑 ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) → (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸)) |
27 | 22, 26 | anim12dan 609 | . . . . 5 ⊢ ((𝜑 ∧ (〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋) ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌))) → ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) |
28 | 1 | adantr 473 | . . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
29 | simprl 758 | . . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸)) | |
30 | simprr 760 | . . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸)) | |
31 | dihopellsm.a | . . . . . . . . 9 ⊢ 𝐴 = (𝑣 ∈ 𝐸, 𝑤 ∈ 𝐸 ↦ (𝑖 ∈ 𝑇 ↦ ((𝑣‘𝑖) ∘ (𝑤‘𝑖)))) | |
32 | 4, 17, 18, 31, 6, 13 | dvhopvadd2 37675 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸)) → (〈𝑔, 𝑡〉(+g‘𝑈)〈ℎ, 𝑢〉) = 〈(𝑔 ∘ ℎ), (𝑡𝐴𝑢)〉) |
33 | 28, 29, 30, 32 | syl3anc 1351 | . . . . . . 7 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (〈𝑔, 𝑡〉(+g‘𝑈)〈ℎ, 𝑢〉) = 〈(𝑔 ∘ ℎ), (𝑡𝐴𝑢)〉) |
34 | 33 | eqeq2d 2782 | . . . . . 6 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (〈𝐹, 𝑆〉 = (〈𝑔, 𝑡〉(+g‘𝑈)〈ℎ, 𝑢〉) ↔ 〈𝐹, 𝑆〉 = 〈(𝑔 ∘ ℎ), (𝑡𝐴𝑢)〉)) |
35 | vex 3412 | . . . . . . . 8 ⊢ 𝑔 ∈ V | |
36 | vex 3412 | . . . . . . . 8 ⊢ ℎ ∈ V | |
37 | 35, 36 | coex 7444 | . . . . . . 7 ⊢ (𝑔 ∘ ℎ) ∈ V |
38 | ovex 7002 | . . . . . . 7 ⊢ (𝑡𝐴𝑢) ∈ V | |
39 | 37, 38 | opth2 5223 | . . . . . 6 ⊢ (〈𝐹, 𝑆〉 = 〈(𝑔 ∘ ℎ), (𝑡𝐴𝑢)〉 ↔ (𝐹 = (𝑔 ∘ ℎ) ∧ 𝑆 = (𝑡𝐴𝑢))) |
40 | 34, 39 | syl6bb 279 | . . . . 5 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (〈𝐹, 𝑆〉 = (〈𝑔, 𝑡〉(+g‘𝑈)〈ℎ, 𝑢〉) ↔ (𝐹 = (𝑔 ∘ ℎ) ∧ 𝑆 = (𝑡𝐴𝑢)))) |
41 | 27, 40 | syldan 582 | . . . 4 ⊢ ((𝜑 ∧ (〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋) ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌))) → (〈𝐹, 𝑆〉 = (〈𝑔, 𝑡〉(+g‘𝑈)〈ℎ, 𝑢〉) ↔ (𝐹 = (𝑔 ∘ ℎ) ∧ 𝑆 = (𝑡𝐴𝑢)))) |
42 | 41 | pm5.32da 571 | . . 3 ⊢ (𝜑 → (((〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋) ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) ∧ 〈𝐹, 𝑆〉 = (〈𝑔, 𝑡〉(+g‘𝑈)〈ℎ, 𝑢〉)) ↔ ((〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋) ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) ∧ (𝐹 = (𝑔 ∘ ℎ) ∧ 𝑆 = (𝑡𝐴𝑢))))) |
43 | 42 | 4exbidv 1885 | . 2 ⊢ (𝜑 → (∃𝑔∃𝑡∃ℎ∃𝑢((〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋) ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) ∧ 〈𝐹, 𝑆〉 = (〈𝑔, 𝑡〉(+g‘𝑈)〈ℎ, 𝑢〉)) ↔ ∃𝑔∃𝑡∃ℎ∃𝑢((〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋) ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) ∧ (𝐹 = (𝑔 ∘ ℎ) ∧ 𝑆 = (𝑡𝐴𝑢))))) |
44 | 16, 43 | bitrd 271 | 1 ⊢ (𝜑 → (〈𝐹, 𝑆〉 ∈ ((𝐼‘𝑋) ⊕ (𝐼‘𝑌)) ↔ ∃𝑔∃𝑡∃ℎ∃𝑢((〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋) ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) ∧ (𝐹 = (𝑔 ∘ ℎ) ∧ 𝑆 = (𝑡𝐴𝑢))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∃wex 1742 ∈ wcel 2050 〈cop 4441 ↦ cmpt 5002 ∘ ccom 5405 ‘cfv 6182 (class class class)co 6970 ∈ cmpo 6972 Basecbs 16333 +gcplusg 16415 LSSumclsm 18514 LSubSpclss 19419 HLchlt 35931 LHypclh 36565 LTrncltrn 36682 TEndoctendo 37333 DVecHcdvh 37659 DIsoHcdih 37809 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 ax-riotaBAD 35534 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7495 df-2nd 7496 df-tpos 7689 df-undef 7736 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-1o 7899 df-oadd 7903 df-er 8083 df-map 8202 df-en 8301 df-dom 8302 df-sdom 8303 df-fin 8304 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-nn 11434 df-2 11497 df-3 11498 df-4 11499 df-5 11500 df-6 11501 df-n0 11702 df-z 11788 df-uz 12053 df-fz 12703 df-struct 16335 df-ndx 16336 df-slot 16337 df-base 16339 df-sets 16340 df-ress 16341 df-plusg 16428 df-mulr 16429 df-sca 16431 df-vsca 16432 df-0g 16565 df-proset 17390 df-poset 17408 df-plt 17420 df-lub 17436 df-glb 17437 df-join 17438 df-meet 17439 df-p0 17501 df-p1 17502 df-lat 17508 df-clat 17570 df-mgm 17704 df-sgrp 17746 df-mnd 17757 df-submnd 17798 df-grp 17888 df-minusg 17889 df-sbg 17890 df-subg 18054 df-cntz 18212 df-lsm 18516 df-cmn 18662 df-abl 18663 df-mgp 18957 df-ur 18969 df-ring 19016 df-oppr 19090 df-dvdsr 19108 df-unit 19109 df-invr 19139 df-dvr 19150 df-drng 19221 df-lmod 19352 df-lss 19420 df-lsp 19460 df-lvec 19591 df-oposet 35757 df-ol 35759 df-oml 35760 df-covers 35847 df-ats 35848 df-atl 35879 df-cvlat 35903 df-hlat 35932 df-llines 36079 df-lplanes 36080 df-lvols 36081 df-lines 36082 df-psubsp 36084 df-pmap 36085 df-padd 36377 df-lhyp 36569 df-laut 36570 df-ldil 36685 df-ltrn 36686 df-trl 36740 df-tendo 37336 df-edring 37338 df-disoa 37610 df-dvech 37660 df-dib 37720 df-dic 37754 df-dih 37810 |
This theorem is referenced by: dihjatcclem4 38002 |
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