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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 2824 | . . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
8 | 3, 4, 5, 6, 7 | dihlss 38390 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ (LSubSp‘𝑈)) |
9 | 1, 2, 8 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (LSubSp‘𝑈)) |
10 | dihopellsm.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
11 | 3, 4, 5, 6, 7 | dihlss 38390 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵) → (𝐼‘𝑌) ∈ (LSubSp‘𝑈)) |
12 | 1, 10, 11 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝐼‘𝑌) ∈ (LSubSp‘𝑈)) |
13 | eqid 2824 | . . . 4 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
14 | dihopellsm.p | . . . 4 ⊢ ⊕ = (LSSum‘𝑈) | |
15 | 4, 6, 13, 7, 14 | dvhopellsm 38257 | . . 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 1367 | . 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 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
20 | 2 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋)) → 𝑋 ∈ 𝐵) |
21 | simpr 487 | . . . . . . 7 ⊢ ((𝜑 ∧ 〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋)) → 〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋)) | |
22 | 3, 4, 17, 18, 5, 19, 20, 21 | dihopcl 38393 | . . . . . 6 ⊢ ((𝜑 ∧ 〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋)) → (𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸)) |
23 | 1 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
24 | 10 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) → 𝑌 ∈ 𝐵) |
25 | simpr 487 | . . . . . . 7 ⊢ ((𝜑 ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) → 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) | |
26 | 3, 4, 17, 18, 5, 23, 24, 25 | dihopcl 38393 | . . . . . 6 ⊢ ((𝜑 ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) → (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸)) |
27 | 22, 26 | anim12dan 620 | . . . . 5 ⊢ ((𝜑 ∧ (〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋) ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌))) → ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) |
28 | 1 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
29 | simprl 769 | . . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸)) | |
30 | simprr 771 | . . . . . . . 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 38234 | . . . . . . . 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 1367 | . . . . . . 7 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (〈𝑔, 𝑡〉(+g‘𝑈)〈ℎ, 𝑢〉) = 〈(𝑔 ∘ ℎ), (𝑡𝐴𝑢)〉) |
34 | 33 | eqeq2d 2835 | . . . . . 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 3500 | . . . . . . . 8 ⊢ 𝑔 ∈ V | |
36 | vex 3500 | . . . . . . . 8 ⊢ ℎ ∈ V | |
37 | 35, 36 | coex 7638 | . . . . . . 7 ⊢ (𝑔 ∘ ℎ) ∈ V |
38 | ovex 7192 | . . . . . . 7 ⊢ (𝑡𝐴𝑢) ∈ V | |
39 | 37, 38 | opth2 5375 | . . . . . 6 ⊢ (〈𝐹, 𝑆〉 = 〈(𝑔 ∘ ℎ), (𝑡𝐴𝑢)〉 ↔ (𝐹 = (𝑔 ∘ ℎ) ∧ 𝑆 = (𝑡𝐴𝑢))) |
40 | 34, 39 | syl6bb 289 | . . . . 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 593 | . . . 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 581 | . . 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 1926 | . 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 281 | 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 208 ∧ wa 398 = wceq 1536 ∃wex 1779 ∈ wcel 2113 〈cop 4576 ↦ cmpt 5149 ∘ ccom 5562 ‘cfv 6358 (class class class)co 7159 ∈ cmpo 7161 Basecbs 16486 +gcplusg 16568 LSSumclsm 18762 LSubSpclss 19706 HLchlt 36490 LHypclh 37124 LTrncltrn 37241 TEndoctendo 37892 DVecHcdvh 38218 DIsoHcdih 38368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-riotaBAD 36093 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-tpos 7895 df-undef 7942 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-sca 16584 df-vsca 16585 df-0g 16718 df-proset 17541 df-poset 17559 df-plt 17571 df-lub 17587 df-glb 17588 df-join 17589 df-meet 17590 df-p0 17652 df-p1 17653 df-lat 17659 df-clat 17721 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-grp 18109 df-minusg 18110 df-sbg 18111 df-subg 18279 df-cntz 18450 df-lsm 18764 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-oppr 19376 df-dvdsr 19394 df-unit 19395 df-invr 19425 df-dvr 19436 df-drng 19507 df-lmod 19639 df-lss 19707 df-lsp 19747 df-lvec 19878 df-oposet 36316 df-ol 36318 df-oml 36319 df-covers 36406 df-ats 36407 df-atl 36438 df-cvlat 36462 df-hlat 36491 df-llines 36638 df-lplanes 36639 df-lvols 36640 df-lines 36641 df-psubsp 36643 df-pmap 36644 df-padd 36936 df-lhyp 37128 df-laut 37129 df-ldil 37244 df-ltrn 37245 df-trl 37299 df-tendo 37895 df-edring 37897 df-disoa 38169 df-dvech 38219 df-dib 38279 df-dic 38313 df-dih 38369 |
This theorem is referenced by: dihjatcclem4 38561 |
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