mapdrvallem3 - Mathbox for Norm Megill

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Theorem mapdrvallem3 41355

Description: Lemma for mapdrval 41356. (Contributed by NM, 2-Feb-2015.)

**Hypotheses**
Ref	Expression
mapdrval.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdrval.o	⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
mapdrval.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdrval.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdrval.s	⊢ 𝑆 = (LSubSp‘𝑈)
mapdrval.f	⊢ 𝐹 = (LFnl‘𝑈)
mapdrval.l	⊢ 𝐿 = (LKer‘𝑈)
mapdrval.d	⊢ 𝐷 = (LDual‘𝑈)
mapdrval.t	⊢ 𝑇 = (LSubSp‘𝐷)
mapdrval.c	⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)}
mapdrval.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
mapdrval.r	⊢ (𝜑 → 𝑅 ∈ 𝑇)
mapdrval.e	⊢ (𝜑 → 𝑅 ⊆ 𝐶)
mapdrval.q	⊢ 𝑄 = ∪ ℎ ∈ 𝑅 (𝑂‘(𝐿‘ℎ))
mapdrval.v	⊢ 𝑉 = (Base‘𝑈)
mapdrvallem2.a	⊢ 𝐴 = (LSAtoms‘𝑈)
mapdrvallem2.n	⊢ 𝑁 = (LSpan‘𝑈)
mapdrvallem2.z	⊢ 0 = (0_g‘𝑈)
mapdrvallem2.y	⊢ 𝑌 = (0_g‘𝐷)

**Assertion**
Ref	Expression
mapdrvallem3	⊢ (𝜑 → {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑄} = 𝑅)

Distinct variable groups: 𝐶,𝑓 𝑓,𝑔,𝐹 𝑓,𝐾 𝑔,ℎ,𝐿 𝑔,𝑂,ℎ 𝑄,𝑓,ℎ 𝑅,𝑓,ℎ 𝑈,𝑔 𝑓,𝑊 𝜑,𝑓 𝐶,ℎ ℎ,𝑁 𝑄,ℎ 𝑈,ℎ ℎ,𝑉 ℎ,𝑌 0 ,ℎ 𝜑,ℎ Allowed substitution hints: 𝜑(𝑔) 𝐴(𝑓,𝑔,ℎ) 𝐶(𝑔) 𝐷(𝑓,𝑔,ℎ) 𝑄(𝑔) 𝑅(𝑔) 𝑆(𝑓,𝑔,ℎ) 𝑇(𝑓,𝑔,ℎ) 𝑈(𝑓) 𝐹(ℎ) 𝐻(𝑓,𝑔,ℎ) 𝐾(𝑔,ℎ) 𝐿(𝑓) 𝑀(𝑓,𝑔,ℎ) 𝑁(𝑓,𝑔) 𝑂(𝑓) 𝑉(𝑓,𝑔) 𝑊(𝑔,ℎ) 𝑌(𝑓,𝑔) 0 (𝑓,𝑔)

**Proof of Theorem mapdrvallem3**
Step	Hyp	Ref	Expression
1		mapdrval.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		mapdrval.o	. . 3 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
3		mapdrval.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
4		mapdrval.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		mapdrval.s	. . 3 ⊢ 𝑆 = (LSubSp‘𝑈)
6		mapdrval.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑈)
7		mapdrval.l	. . 3 ⊢ 𝐿 = (LKer‘𝑈)
8		mapdrval.d	. . 3 ⊢ 𝐷 = (LDual‘𝑈)
9		mapdrval.t	. . 3 ⊢ 𝑇 = (LSubSp‘𝐷)
10		mapdrval.c	. . 3 ⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)}
11		mapdrval.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
12		mapdrval.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑇)
13		mapdrval.e	. . 3 ⊢ (𝜑 → 𝑅 ⊆ 𝐶)
14		mapdrval.q	. . 3 ⊢ 𝑄 = ∪ ℎ ∈ 𝑅 (𝑂‘(𝐿‘ℎ))
15		mapdrval.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
16		mapdrvallem2.a	. . 3 ⊢ 𝐴 = (LSAtoms‘𝑈)
17		mapdrvallem2.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
18		mapdrvallem2.z	. . 3 ⊢ 0 = (0_g‘𝑈)
19		mapdrvallem2.y	. . 3 ⊢ 𝑌 = (0_g‘𝐷)
20	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19	mapdrvallem2 41354	. 2 ⊢ (𝜑 → {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑄} ⊆ 𝑅)
21		2fveq3 6895	. . . . . 6 ⊢ (ℎ = 𝑓 → (𝑂‘(𝐿‘ℎ)) = (𝑂‘(𝐿‘𝑓)))
22	21	ssiun2s 5048	. . . . 5 ⊢ (𝑓 ∈ 𝑅 → (𝑂‘(𝐿‘𝑓)) ⊆ ∪ ℎ ∈ 𝑅 (𝑂‘(𝐿‘ℎ)))
23	22	adantl 480	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑅) → (𝑂‘(𝐿‘𝑓)) ⊆ ∪ ℎ ∈ 𝑅 (𝑂‘(𝐿‘ℎ)))
24	23, 14	sseqtrrdi 4030	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑅) → (𝑂‘(𝐿‘𝑓)) ⊆ 𝑄)
25	13, 24	ssrabdv 4067	. 2 ⊢ (𝜑 → 𝑅 ⊆ {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑄})
26	20, 25	eqssd 3996	1 ⊢ (𝜑 → {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑄} = 𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 {crab 3419 ⊆ wss 3946 ∪ ciun 4993 ‘cfv 6543 Basecbs 17205 0_gc0g 17446 LSubSpclss 20901 LSpanclspn 20941 LSAtomsclsa 38682 LFnlclfn 38765 LKerclk 38793 LDualcld 38831 HLchlt 39058 LHypclh 39693 DVecHcdvh 40787 ocHcoch 41056 mapdcmpd 41333

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-riotaBAD 38661

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-om 7866 df-1st 7992 df-2nd 7993 df-tpos 8230 df-undef 8277 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12256 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-n0 12516 df-z 12602 df-uz 12866 df-fz 13530 df-struct 17141 df-sets 17158 df-slot 17176 df-ndx 17188 df-base 17206 df-ress 17235 df-plusg 17271 df-mulr 17272 df-sca 17274 df-vsca 17275 df-0g 17448 df-proset 18312 df-poset 18330 df-plt 18347 df-lub 18363 df-glb 18364 df-join 18365 df-meet 18366 df-p0 18442 df-p1 18443 df-lat 18449 df-clat 18516 df-mgm 18625 df-sgrp 18704 df-mnd 18720 df-submnd 18766 df-grp 18923 df-minusg 18924 df-sbg 18925 df-subg 19110 df-cntz 19304 df-lsm 19627 df-cmn 19773 df-abl 19774 df-mgp 20111 df-rng 20129 df-ur 20158 df-ring 20211 df-oppr 20309 df-dvdsr 20332 df-unit 20333 df-invr 20363 df-dvr 20376 df-drng 20702 df-lmod 20831 df-lss 20902 df-lsp 20942 df-lvec 21074 df-lsatoms 38684 df-lshyp 38685 df-lfl 38766 df-lkr 38794 df-ldual 38832 df-oposet 38884 df-ol 38886 df-oml 38887 df-covers 38974 df-ats 38975 df-atl 39006 df-cvlat 39030 df-hlat 39059 df-llines 39207 df-lplanes 39208 df-lvols 39209 df-lines 39210 df-psubsp 39212 df-pmap 39213 df-padd 39505 df-lhyp 39697 df-laut 39698 df-ldil 39813 df-ltrn 39814 df-trl 39868 df-tgrp 40452 df-tendo 40464 df-edring 40466 df-dveca 40712 df-disoa 40738 df-dvech 40788 df-dib 40848 df-dic 40882 df-dih 40938 df-doch 41057 df-djh 41104

This theorem is referenced by: mapdrval 41356

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