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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xihopellsmN | Structured version Visualization version GIF version |
Description: Ordered pair membership in a subspace sum of isomorphism H values. (Contributed by NM, 26-Sep-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
xihopellsm.b | ⊢ 𝐵 = (Base‘𝐾) |
xihopellsm.h | ⊢ 𝐻 = (LHyp‘𝐾) |
xihopellsm.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
xihopellsm.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
xihopellsm.a | ⊢ 𝐴 = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))) |
xihopellsm.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
xihopellsm.l | ⊢ 𝐿 = (LSubSp‘𝑈) |
xihopellsm.p | ⊢ ⊕ = (LSSum‘𝑈) |
xihopellsm.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
xihopellsm.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
xihopellsm.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
xihopellsm.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
xihopellsmN | ⊢ (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 | xihopellsm.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | xihopellsm.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | xihopellsm.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
4 | xihopellsm.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | xihopellsm.i | . . . . 5 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
6 | xihopellsm.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
7 | eqid 2793 | . . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
8 | 3, 4, 5, 6, 7 | dihlss 37867 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ (LSubSp‘𝑈)) |
9 | 1, 2, 8 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (LSubSp‘𝑈)) |
10 | xihopellsm.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
11 | 3, 4, 5, 6, 7 | dihlss 37867 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵) → (𝐼‘𝑌) ∈ (LSubSp‘𝑈)) |
12 | 1, 10, 11 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐼‘𝑌) ∈ (LSubSp‘𝑈)) |
13 | eqid 2793 | . . . 4 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
14 | xihopellsm.p | . . . 4 ⊢ ⊕ = (LSSum‘𝑈) | |
15 | 4, 6, 13, 7, 14 | dvhopellsm 37734 | . . 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 1362 | . 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 | xihopellsm.t | . . . . . . 7 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
18 | xihopellsm.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 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋)) → 〈𝑔, 𝑡〉 ∈ (𝐼‘𝑋)) | |
22 | 3, 4, 17, 18, 5, 19, 20, 21 | dihopcl 37870 | . . . . . 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 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) → 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) | |
26 | 3, 4, 17, 18, 5, 23, 24, 25 | dihopcl 37870 | . . . . . 6 ⊢ ((𝜑 ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑌)) → (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸)) |
27 | 22, 26 | anim12dan 618 | . . . . 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 767 | . . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸)) | |
30 | simprr 769 | . . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸)) | |
31 | xihopellsm.a | . . . . . . . . 9 ⊢ 𝐴 = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))) | |
32 | 4, 17, 18, 31, 6, 13 | dvhopvadd2 37711 | . . . . . . . 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 1362 | . . . . . . 7 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (〈𝑔, 𝑡〉(+g‘𝑈)〈ℎ, 𝑢〉) = 〈(𝑔 ∘ ℎ), (𝑡𝐴𝑢)〉) |
34 | 33 | eqeq2d 2803 | . . . . . 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 3435 | . . . . . . . 8 ⊢ 𝑔 ∈ V | |
36 | vex 3435 | . . . . . . . 8 ⊢ ℎ ∈ V | |
37 | 35, 36 | coex 7482 | . . . . . . 7 ⊢ (𝑔 ∘ ℎ) ∈ V |
38 | ovex 7039 | . . . . . . 7 ⊢ (𝑡𝐴𝑢) ∈ V | |
39 | 37, 38 | opth2 5257 | . . . . . 6 ⊢ (〈𝐹, 𝑆〉 = 〈(𝑔 ∘ ℎ), (𝑡𝐴𝑢)〉 ↔ (𝐹 = (𝑔 ∘ ℎ) ∧ 𝑆 = (𝑡𝐴𝑢))) |
40 | 34, 39 | syl6bb 288 | . . . . 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 591 | . . . 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 579 | . . 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 1902 | . 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 280 | 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 207 ∧ wa 396 = wceq 1520 ∃wex 1759 ∈ wcel 2079 〈cop 4472 ↦ cmpt 5035 ∘ ccom 5439 ‘cfv 6217 (class class class)co 7007 ∈ cmpo 7009 Basecbs 16300 +gcplusg 16382 LSSumclsm 18477 LSubSpclss 19381 HLchlt 35967 LHypclh 36601 LTrncltrn 36718 TEndoctendo 37369 DVecHcdvh 37695 DIsoHcdih 37845 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 ax-riotaBAD 35570 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-fal 1533 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-iin 4822 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-1st 7536 df-2nd 7537 df-tpos 7734 df-undef 7781 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-oadd 7948 df-er 8130 df-map 8249 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-nn 11476 df-2 11537 df-3 11538 df-4 11539 df-5 11540 df-6 11541 df-n0 11735 df-z 11819 df-uz 12083 df-fz 12732 df-struct 16302 df-ndx 16303 df-slot 16304 df-base 16306 df-sets 16307 df-ress 16308 df-plusg 16395 df-mulr 16396 df-sca 16398 df-vsca 16399 df-0g 16532 df-proset 17355 df-poset 17373 df-plt 17385 df-lub 17401 df-glb 17402 df-join 17403 df-meet 17404 df-p0 17466 df-p1 17467 df-lat 17473 df-clat 17535 df-mgm 17669 df-sgrp 17711 df-mnd 17722 df-submnd 17763 df-grp 17852 df-minusg 17853 df-sbg 17854 df-subg 18018 df-cntz 18176 df-lsm 18479 df-cmn 18623 df-abl 18624 df-mgp 18918 df-ur 18930 df-ring 18977 df-oppr 19051 df-dvdsr 19069 df-unit 19070 df-invr 19100 df-dvr 19111 df-drng 19182 df-lmod 19314 df-lss 19382 df-lsp 19422 df-lvec 19553 df-oposet 35793 df-ol 35795 df-oml 35796 df-covers 35883 df-ats 35884 df-atl 35915 df-cvlat 35939 df-hlat 35968 df-llines 36115 df-lplanes 36116 df-lvols 36117 df-lines 36118 df-psubsp 36120 df-pmap 36121 df-padd 36413 df-lhyp 36605 df-laut 36606 df-ldil 36721 df-ltrn 36722 df-trl 36776 df-tendo 37372 df-edring 37374 df-disoa 37646 df-dvech 37696 df-dib 37756 df-dic 37790 df-dih 37846 |
This theorem is referenced by: (None) |
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