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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 2622 | . . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
8 | 3, 4, 5, 6, 7 | dihlss 36539 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ (LSubSp‘𝑈)) |
9 | 1, 2, 8 | syl2anc 693 | . . 3 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (LSubSp‘𝑈)) |
10 | dihopellsm.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
11 | 3, 4, 5, 6, 7 | dihlss 36539 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵) → (𝐼‘𝑌) ∈ (LSubSp‘𝑈)) |
12 | 1, 10, 11 | syl2anc 693 | . . 3 ⊢ (𝜑 → (𝐼‘𝑌) ∈ (LSubSp‘𝑈)) |
13 | eqid 2622 | . . . 4 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
14 | dihopellsm.p | . . . 4 ⊢ ⊕ = (LSSum‘𝑈) | |
15 | 4, 6, 13, 7, 14 | dvhopellsm 36406 | . . 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 1326 | . 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 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
20 | 2 | adantr 481 | . . . . . . 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 36542 | . . . . . 6 ⊢ ((𝜑 ∧ 〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋)) → (𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸)) |
23 | 1 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
24 | 10 | adantr 481 | . . . . . . 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 36542 | . . . . . 6 ⊢ ((𝜑 ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) → (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸)) |
27 | 22, 26 | anim12dan 882 | . . . . 5 ⊢ ((𝜑 ∧ (〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋) ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌))) → ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) |
28 | 1 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
29 | simprl 794 | . . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸)) | |
30 | simprr 796 | . . . . . . . 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 36383 | . . . . . . . 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 1326 | . . . . . . 7 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (〈𝑔, 𝑡〉(+g‘𝑈)〈ℎ, 𝑢〉) = 〈(𝑔 ∘ ℎ), (𝑡𝐴𝑢)〉) |
34 | 33 | eqeq2d 2632 | . . . . . 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 3203 | . . . . . . . 8 ⊢ 𝑔 ∈ V | |
36 | vex 3203 | . . . . . . . 8 ⊢ ℎ ∈ V | |
37 | 35, 36 | coex 7118 | . . . . . . 7 ⊢ (𝑔 ∘ ℎ) ∈ V |
38 | ovex 6678 | . . . . . . 7 ⊢ (𝑡𝐴𝑢) ∈ V | |
39 | 37, 38 | opth2 4949 | . . . . . 6 ⊢ (〈𝐹, 𝑆〉 = 〈(𝑔 ∘ ℎ), (𝑡𝐴𝑢)〉 ↔ (𝐹 = (𝑔 ∘ ℎ) ∧ 𝑆 = (𝑡𝐴𝑢))) |
40 | 34, 39 | syl6bb 276 | . . . . 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 487 | . . . 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 673 | . . 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 1854 | . 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 268 | 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 196 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 〈cop 4183 ↦ cmpt 4729 ∘ ccom 5118 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 Basecbs 15857 +gcplusg 15941 LSSumclsm 18049 LSubSpclss 18932 HLchlt 34637 LHypclh 35270 LTrncltrn 35387 TEndoctendo 36040 DVecHcdvh 36367 DIsoHcdih 36517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-riotaBAD 34239 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-undef 7399 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-0g 16102 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cntz 17750 df-lsm 18051 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lvols 34786 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446 df-tendo 36043 df-edring 36045 df-disoa 36318 df-dvech 36368 df-dib 36428 df-dic 36462 df-dih 36518 |
This theorem is referenced by: dihjatcclem4 36710 |
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