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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 2733 | . . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
8 | 3, 4, 5, 6, 7 | dihlss 40059 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ (LSubSp‘𝑈)) |
9 | 1, 2, 8 | syl2anc 585 | . . 3 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (LSubSp‘𝑈)) |
10 | dihopellsm.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
11 | 3, 4, 5, 6, 7 | dihlss 40059 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵) → (𝐼‘𝑌) ∈ (LSubSp‘𝑈)) |
12 | 1, 10, 11 | syl2anc 585 | . . 3 ⊢ (𝜑 → (𝐼‘𝑌) ∈ (LSubSp‘𝑈)) |
13 | eqid 2733 | . . . 4 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
14 | dihopellsm.p | . . . 4 ⊢ ⊕ = (LSSum‘𝑈) | |
15 | 4, 6, 13, 7, 14 | dvhopellsm 39926 | . . 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 1372 | . 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 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
20 | 2 | adantr 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋)) → 𝑋 ∈ 𝐵) |
21 | simpr 486 | . . . . . . 7 ⊢ ((𝜑 ∧ 〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋)) → 〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋)) | |
22 | 3, 4, 17, 18, 5, 19, 20, 21 | dihopcl 40062 | . . . . . 6 ⊢ ((𝜑 ∧ 〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋)) → (𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸)) |
23 | 1 | adantr 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
24 | 10 | adantr 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) → 𝑌 ∈ 𝐵) |
25 | simpr 486 | . . . . . . 7 ⊢ ((𝜑 ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) → 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) | |
26 | 3, 4, 17, 18, 5, 23, 24, 25 | dihopcl 40062 | . . . . . 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 482 | . . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
29 | simprl 770 | . . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸)) | |
30 | simprr 772 | . . . . . . . 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 39903 | . . . . . . . 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 1372 | . . . . . . 7 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (〈𝑔, 𝑡〉(+g‘𝑈)〈ℎ, 𝑢〉) = 〈(𝑔 ∘ ℎ), (𝑡𝐴𝑢)〉) |
34 | 33 | eqeq2d 2744 | . . . . . 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 3479 | . . . . . . . 8 ⊢ 𝑔 ∈ V | |
36 | vex 3479 | . . . . . . . 8 ⊢ ℎ ∈ V | |
37 | 35, 36 | coex 7916 | . . . . . . 7 ⊢ (𝑔 ∘ ℎ) ∈ V |
38 | ovex 7437 | . . . . . . 7 ⊢ (𝑡𝐴𝑢) ∈ V | |
39 | 37, 38 | opth2 5479 | . . . . . 6 ⊢ (〈𝐹, 𝑆〉 = 〈(𝑔 ∘ ℎ), (𝑡𝐴𝑢)〉 ↔ (𝐹 = (𝑔 ∘ ℎ) ∧ 𝑆 = (𝑡𝐴𝑢))) |
40 | 34, 39 | bitrdi 287 | . . . . 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 592 | . . . 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 580 | . . 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 1930 | . 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 279 | 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 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 〈cop 4633 ↦ cmpt 5230 ∘ ccom 5679 ‘cfv 6540 (class class class)co 7404 ∈ cmpo 7406 Basecbs 17140 +gcplusg 17193 LSSumclsm 19495 LSubSpclss 20530 HLchlt 38158 LHypclh 38793 LTrncltrn 38910 TEndoctendo 39561 DVecHcdvh 39887 DIsoHcdih 40037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-riotaBAD 37761 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-tpos 8206 df-undef 8253 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-0g 17383 df-proset 18244 df-poset 18262 df-plt 18279 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-p1 18375 df-lat 18381 df-clat 18448 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-cntz 19175 df-lsm 19497 df-cmn 19643 df-abl 19644 df-mgp 19980 df-ur 19997 df-ring 20049 df-oppr 20139 df-dvdsr 20160 df-unit 20161 df-invr 20191 df-dvr 20204 df-drng 20306 df-lmod 20461 df-lss 20531 df-lsp 20571 df-lvec 20702 df-oposet 37984 df-ol 37986 df-oml 37987 df-covers 38074 df-ats 38075 df-atl 38106 df-cvlat 38130 df-hlat 38159 df-llines 38307 df-lplanes 38308 df-lvols 38309 df-lines 38310 df-psubsp 38312 df-pmap 38313 df-padd 38605 df-lhyp 38797 df-laut 38798 df-ldil 38913 df-ltrn 38914 df-trl 38968 df-tendo 39564 df-edring 39566 df-disoa 39838 df-dvech 39888 df-dib 39948 df-dic 39982 df-dih 40038 |
This theorem is referenced by: dihjatcclem4 40230 |
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