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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdunirnN | Structured version Visualization version GIF version |
Description: Union of the range of the map defined by df-mapd 38243. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mapdrn.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdrn.o | ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) |
mapdrn.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdrn.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdrn.f | ⊢ 𝐹 = (LFnl‘𝑈) |
mapdrn.l | ⊢ 𝐿 = (LKer‘𝑈) |
mapdunirn.c | ⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)} |
mapdunirn.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
Ref | Expression |
---|---|
mapdunirnN | ⊢ (𝜑 → ∪ ran 𝑀 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdrn.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | mapdrn.o | . . . 4 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) | |
3 | mapdrn.m | . . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
4 | mapdrn.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
5 | mapdrn.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑈) | |
6 | mapdrn.l | . . . 4 ⊢ 𝐿 = (LKer‘𝑈) | |
7 | eqid 2771 | . . . 4 ⊢ (LDual‘𝑈) = (LDual‘𝑈) | |
8 | eqid 2771 | . . . 4 ⊢ (LSubSp‘(LDual‘𝑈)) = (LSubSp‘(LDual‘𝑈)) | |
9 | mapdunirn.c | . . . 4 ⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)} | |
10 | mapdunirn.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | mapdrn 38267 | . . 3 ⊢ (𝜑 → ran 𝑀 = ((LSubSp‘(LDual‘𝑈)) ∩ 𝒫 𝐶)) |
12 | 11 | unieqd 4718 | . 2 ⊢ (𝜑 → ∪ ran 𝑀 = ∪ ((LSubSp‘(LDual‘𝑈)) ∩ 𝒫 𝐶)) |
13 | uniin 4728 | . . . 4 ⊢ ∪ ((LSubSp‘(LDual‘𝑈)) ∩ 𝒫 𝐶) ⊆ (∪ (LSubSp‘(LDual‘𝑈)) ∩ ∪ 𝒫 𝐶) | |
14 | eqid 2771 | . . . . . . . 8 ⊢ (Base‘(LDual‘𝑈)) = (Base‘(LDual‘𝑈)) | |
15 | 1, 4, 10 | dvhlmod 37728 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod) |
16 | 7, 15 | lduallmod 35771 | . . . . . . . 8 ⊢ (𝜑 → (LDual‘𝑈) ∈ LMod) |
17 | 14, 8, 16 | lssuni 19445 | . . . . . . 7 ⊢ (𝜑 → ∪ (LSubSp‘(LDual‘𝑈)) = (Base‘(LDual‘𝑈))) |
18 | 5, 7, 14, 15 | ldualvbase 35744 | . . . . . . 7 ⊢ (𝜑 → (Base‘(LDual‘𝑈)) = 𝐹) |
19 | 17, 18 | eqtrd 2807 | . . . . . 6 ⊢ (𝜑 → ∪ (LSubSp‘(LDual‘𝑈)) = 𝐹) |
20 | unipw 5195 | . . . . . . 7 ⊢ ∪ 𝒫 𝐶 = 𝐶 | |
21 | 20 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ∪ 𝒫 𝐶 = 𝐶) |
22 | 19, 21 | ineq12d 4071 | . . . . 5 ⊢ (𝜑 → (∪ (LSubSp‘(LDual‘𝑈)) ∩ ∪ 𝒫 𝐶) = (𝐹 ∩ 𝐶)) |
23 | ssrab2 3939 | . . . . . . . 8 ⊢ {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)} ⊆ 𝐹 | |
24 | 9, 23 | eqsstri 3884 | . . . . . . 7 ⊢ 𝐶 ⊆ 𝐹 |
25 | sseqin2 4073 | . . . . . . 7 ⊢ (𝐶 ⊆ 𝐹 ↔ (𝐹 ∩ 𝐶) = 𝐶) | |
26 | 24, 25 | mpbi 222 | . . . . . 6 ⊢ (𝐹 ∩ 𝐶) = 𝐶 |
27 | 26 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝐹 ∩ 𝐶) = 𝐶) |
28 | 22, 27 | eqtrd 2807 | . . . 4 ⊢ (𝜑 → (∪ (LSubSp‘(LDual‘𝑈)) ∩ ∪ 𝒫 𝐶) = 𝐶) |
29 | 13, 28 | syl5sseq 3902 | . . 3 ⊢ (𝜑 → ∪ ((LSubSp‘(LDual‘𝑈)) ∩ 𝒫 𝐶) ⊆ 𝐶) |
30 | 1, 4, 2, 5, 6, 7, 8, 9, 10 | lclkr 38151 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ (LSubSp‘(LDual‘𝑈))) |
31 | 5 | fvexi 6510 | . . . . . . . 8 ⊢ 𝐹 ∈ V |
32 | 9, 31 | rabex2 5089 | . . . . . . 7 ⊢ 𝐶 ∈ V |
33 | 32 | pwid 4432 | . . . . . 6 ⊢ 𝐶 ∈ 𝒫 𝐶 |
34 | 33 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝒫 𝐶) |
35 | 30, 34 | elind 4053 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ((LSubSp‘(LDual‘𝑈)) ∩ 𝒫 𝐶)) |
36 | elssuni 4737 | . . . 4 ⊢ (𝐶 ∈ ((LSubSp‘(LDual‘𝑈)) ∩ 𝒫 𝐶) → 𝐶 ⊆ ∪ ((LSubSp‘(LDual‘𝑈)) ∩ 𝒫 𝐶)) | |
37 | 35, 36 | syl 17 | . . 3 ⊢ (𝜑 → 𝐶 ⊆ ∪ ((LSubSp‘(LDual‘𝑈)) ∩ 𝒫 𝐶)) |
38 | 29, 37 | eqssd 3868 | . 2 ⊢ (𝜑 → ∪ ((LSubSp‘(LDual‘𝑈)) ∩ 𝒫 𝐶) = 𝐶) |
39 | 12, 38 | eqtrd 2807 | 1 ⊢ (𝜑 → ∪ ran 𝑀 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 {crab 3085 ∩ cin 3821 ⊆ wss 3822 𝒫 cpw 4416 ∪ cuni 4708 ran crn 5404 ‘cfv 6185 Basecbs 16337 LModclmod 19368 LSubSpclss 19437 LFnlclfn 35675 LKerclk 35703 LDualcld 35741 HLchlt 35968 LHypclh 36602 DVecHcdvh 37696 ocHcoch 37965 mapdcmpd 38242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-riotaBAD 35571 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-iin 4791 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-of 7225 df-om 7395 df-1st 7499 df-2nd 7500 df-tpos 7693 df-undef 7740 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-map 8206 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-n0 11706 df-z 11792 df-uz 12057 df-fz 12707 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-sca 16435 df-vsca 16436 df-0g 16569 df-mre 16727 df-mrc 16728 df-acs 16730 df-proset 17408 df-poset 17426 df-plt 17438 df-lub 17454 df-glb 17455 df-join 17456 df-meet 17457 df-p0 17519 df-p1 17520 df-lat 17526 df-clat 17588 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-submnd 17816 df-grp 17906 df-minusg 17907 df-sbg 17908 df-subg 18072 df-cntz 18230 df-oppg 18257 df-lsm 18534 df-cmn 18680 df-abl 18681 df-mgp 18975 df-ur 18987 df-ring 19034 df-oppr 19108 df-dvdsr 19126 df-unit 19127 df-invr 19157 df-dvr 19168 df-drng 19239 df-lmod 19370 df-lss 19438 df-lsp 19478 df-lvec 19609 df-lsatoms 35594 df-lshyp 35595 df-lcv 35637 df-lfl 35676 df-lkr 35704 df-ldual 35742 df-oposet 35794 df-ol 35796 df-oml 35797 df-covers 35884 df-ats 35885 df-atl 35916 df-cvlat 35940 df-hlat 35969 df-llines 36116 df-lplanes 36117 df-lvols 36118 df-lines 36119 df-psubsp 36121 df-pmap 36122 df-padd 36414 df-lhyp 36606 df-laut 36607 df-ldil 36722 df-ltrn 36723 df-trl 36777 df-tgrp 37361 df-tendo 37373 df-edring 37375 df-dveca 37621 df-disoa 37647 df-dvech 37697 df-dib 37757 df-dic 37791 df-dih 37847 df-doch 37966 df-djh 38013 df-mapd 38243 |
This theorem is referenced by: (None) |
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