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 35932. (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 2610 | . . . 4 ⊢ (LDual‘𝑈) = (LDual‘𝑈) | |
8 | eqid 2610 | . . . 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 35956 | . . 3 ⊢ (𝜑 → ran 𝑀 = ((LSubSp‘(LDual‘𝑈)) ∩ 𝒫 𝐶)) |
12 | 11 | unieqd 4382 | . 2 ⊢ (𝜑 → ∪ ran 𝑀 = ∪ ((LSubSp‘(LDual‘𝑈)) ∩ 𝒫 𝐶)) |
13 | uniin 4393 | . . . 4 ⊢ ∪ ((LSubSp‘(LDual‘𝑈)) ∩ 𝒫 𝐶) ⊆ (∪ (LSubSp‘(LDual‘𝑈)) ∩ ∪ 𝒫 𝐶) | |
14 | eqid 2610 | . . . . . . . 8 ⊢ (Base‘(LDual‘𝑈)) = (Base‘(LDual‘𝑈)) | |
15 | 1, 4, 10 | dvhlmod 35417 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod) |
16 | 7, 15 | lduallmod 33458 | . . . . . . . 8 ⊢ (𝜑 → (LDual‘𝑈) ∈ LMod) |
17 | 14, 8, 16 | lssuni 18761 | . . . . . . 7 ⊢ (𝜑 → ∪ (LSubSp‘(LDual‘𝑈)) = (Base‘(LDual‘𝑈))) |
18 | 5, 7, 14, 15 | ldualvbase 33431 | . . . . . . 7 ⊢ (𝜑 → (Base‘(LDual‘𝑈)) = 𝐹) |
19 | 17, 18 | eqtrd 2644 | . . . . . 6 ⊢ (𝜑 → ∪ (LSubSp‘(LDual‘𝑈)) = 𝐹) |
20 | unipw 4845 | . . . . . . 7 ⊢ ∪ 𝒫 𝐶 = 𝐶 | |
21 | 20 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ∪ 𝒫 𝐶 = 𝐶) |
22 | 19, 21 | ineq12d 3777 | . . . . 5 ⊢ (𝜑 → (∪ (LSubSp‘(LDual‘𝑈)) ∩ ∪ 𝒫 𝐶) = (𝐹 ∩ 𝐶)) |
23 | ssrab2 3650 | . . . . . . . 8 ⊢ {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)} ⊆ 𝐹 | |
24 | 9, 23 | eqsstri 3598 | . . . . . . 7 ⊢ 𝐶 ⊆ 𝐹 |
25 | sseqin2 3779 | . . . . . . 7 ⊢ (𝐶 ⊆ 𝐹 ↔ (𝐹 ∩ 𝐶) = 𝐶) | |
26 | 24, 25 | mpbi 219 | . . . . . 6 ⊢ (𝐹 ∩ 𝐶) = 𝐶 |
27 | 26 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝐹 ∩ 𝐶) = 𝐶) |
28 | 22, 27 | eqtrd 2644 | . . . 4 ⊢ (𝜑 → (∪ (LSubSp‘(LDual‘𝑈)) ∩ ∪ 𝒫 𝐶) = 𝐶) |
29 | 13, 28 | syl5sseq 3616 | . . 3 ⊢ (𝜑 → ∪ ((LSubSp‘(LDual‘𝑈)) ∩ 𝒫 𝐶) ⊆ 𝐶) |
30 | 1, 4, 2, 5, 6, 7, 8, 9, 10 | lclkr 35840 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ (LSubSp‘(LDual‘𝑈))) |
31 | fvex 6113 | . . . . . . . . 9 ⊢ (LFnl‘𝑈) ∈ V | |
32 | 5, 31 | eqeltri 2684 | . . . . . . . 8 ⊢ 𝐹 ∈ V |
33 | 9, 32 | rabex2 4742 | . . . . . . 7 ⊢ 𝐶 ∈ V |
34 | 33 | pwid 4122 | . . . . . 6 ⊢ 𝐶 ∈ 𝒫 𝐶 |
35 | 34 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝒫 𝐶) |
36 | 30, 35 | elind 3760 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ((LSubSp‘(LDual‘𝑈)) ∩ 𝒫 𝐶)) |
37 | elssuni 4403 | . . . 4 ⊢ (𝐶 ∈ ((LSubSp‘(LDual‘𝑈)) ∩ 𝒫 𝐶) → 𝐶 ⊆ ∪ ((LSubSp‘(LDual‘𝑈)) ∩ 𝒫 𝐶)) | |
38 | 36, 37 | syl 17 | . . 3 ⊢ (𝜑 → 𝐶 ⊆ ∪ ((LSubSp‘(LDual‘𝑈)) ∩ 𝒫 𝐶)) |
39 | 29, 38 | eqssd 3585 | . 2 ⊢ (𝜑 → ∪ ((LSubSp‘(LDual‘𝑈)) ∩ 𝒫 𝐶) = 𝐶) |
40 | 12, 39 | eqtrd 2644 | 1 ⊢ (𝜑 → ∪ ran 𝑀 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 𝒫 cpw 4108 ∪ cuni 4372 ran crn 5039 ‘cfv 5804 Basecbs 15695 LModclmod 18686 LSubSpclss 18753 LFnlclfn 33362 LKerclk 33390 LDualcld 33428 HLchlt 33655 LHypclh 34288 DVecHcdvh 35385 ocHcoch 35654 mapdcmpd 35931 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-riotaBAD 33257 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-undef 7286 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-0g 15925 df-mre 16069 df-mrc 16070 df-acs 16072 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-cntz 17573 df-oppg 17599 df-lsm 17874 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-drng 18572 df-lmod 18688 df-lss 18754 df-lsp 18793 df-lvec 18924 df-lsatoms 33281 df-lshyp 33282 df-lcv 33324 df-lfl 33363 df-lkr 33391 df-ldual 33429 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-llines 33802 df-lplanes 33803 df-lvols 33804 df-lines 33805 df-psubsp 33807 df-pmap 33808 df-padd 34100 df-lhyp 34292 df-laut 34293 df-ldil 34408 df-ltrn 34409 df-trl 34464 df-tgrp 35049 df-tendo 35061 df-edring 35063 df-dveca 35309 df-disoa 35336 df-dvech 35386 df-dib 35446 df-dic 35480 df-dih 35536 df-doch 35655 df-djh 35702 df-mapd 35932 |
This theorem is referenced by: (None) |
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