dih1dimb2 - Mathbox for Norm Megill

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Theorem dih1dimb2 41705

Description: Isomorphism H at an atom under 𝑊. (Contributed by NM, 27-Apr-2014.)

**Hypotheses**
Ref	Expression
dih1dimb2.b	⊢ 𝐵 = (Base‘𝐾)
dih1dimb2.l	⊢ ≤ = (le‘𝐾)
dih1dimb2.a	⊢ 𝐴 = (Atoms‘𝐾)
dih1dimb2.h	⊢ 𝐻 = (LHyp‘𝐾)
dih1dimb2.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dih1dimb2.o	⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
dih1dimb2.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dih1dimb2.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dih1dimb2.n	⊢ 𝑁 = (LSpan‘𝑈)

**Assertion**
Ref	Expression
dih1dimb2	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝐼‘𝑄) = (𝑁‘{⟨𝑓, 𝑂⟩})))

Distinct variable groups: ≤ ,𝑓 𝐴,𝑓 𝐵,ℎ 𝑓,𝐻 𝑓,ℎ,𝐾 𝑄,𝑓 𝑇,𝑓,ℎ 𝑓,𝑊,ℎ Allowed substitution hints: 𝐴(ℎ) 𝐵(𝑓) 𝑄(ℎ) 𝑈(𝑓,ℎ) 𝐻(ℎ) 𝐼(𝑓,ℎ) ≤ (ℎ) 𝑁(𝑓,ℎ) 𝑂(𝑓,ℎ)

**Proof of Theorem dih1dimb2**
Step	Hyp	Ref	Expression
1		dih1dimb2.l	. . 3 ⊢ ≤ = (le‘𝐾)
2		dih1dimb2.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
3		dih1dimb2.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
4		dih1dimb2.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
5		eqid 2737	. . 3 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	cdlemf 41027	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄)
7		simp3 1139	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄)
8		simp1rl 1240	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → 𝑄 ∈ 𝐴)
9	7, 8	eqeltrd 2837	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (((trL‘𝐾)‘𝑊)‘𝑓) ∈ 𝐴)
10		simp1l 1199	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11		simp2 1138	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → 𝑓 ∈ 𝑇)
12		dih1dimb2.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾)
13	12, 2, 3, 4, 5	trlnidatb 40641	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ 𝑇) → (𝑓 ≠ ( I ↾ 𝐵) ↔ (((trL‘𝐾)‘𝑊)‘𝑓) ∈ 𝐴))
14	10, 11, 13	syl2anc 585	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (𝑓 ≠ ( I ↾ 𝐵) ↔ (((trL‘𝐾)‘𝑊)‘𝑓) ∈ 𝐴))
15	9, 14	mpbird 257	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → 𝑓 ≠ ( I ↾ 𝐵))
16	7	fveq2d 6840	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (𝐼‘(((trL‘𝐾)‘𝑊)‘𝑓)) = (𝐼‘𝑄))
17		dih1dimb2.o	. . . . . . . 8 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
18		dih1dimb2.u	. . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
19		dih1dimb2.i	. . . . . . . 8 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
20		dih1dimb2.n	. . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑈)
21	12, 3, 4, 5, 17, 18, 19, 20	dih1dimb 41704	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ 𝑇) → (𝐼‘(((trL‘𝐾)‘𝑊)‘𝑓)) = (𝑁‘{⟨𝑓, 𝑂⟩}))
22	10, 11, 21	syl2anc 585	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (𝐼‘(((trL‘𝐾)‘𝑊)‘𝑓)) = (𝑁‘{⟨𝑓, 𝑂⟩}))
23	16, 22	eqtr3d 2774	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (𝐼‘𝑄) = (𝑁‘{⟨𝑓, 𝑂⟩}))
24	15, 23	jca 511	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝐼‘𝑄) = (𝑁‘{⟨𝑓, 𝑂⟩})))
25	24	3expia 1122	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) → ((((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄 → (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝐼‘𝑄) = (𝑁‘{⟨𝑓, 𝑂⟩}))))
26	25	reximdva 3151	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → (∃𝑓 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄 → ∃𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝐼‘𝑄) = (𝑁‘{⟨𝑓, 𝑂⟩}))))
27	6, 26	mpd 15	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝐼‘𝑄) = (𝑁‘{⟨𝑓, 𝑂⟩})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∃wrex 3062 {csn 4568 ⟨cop 4574 class class class wbr 5086 ↦ cmpt 5167 I cid 5520 ↾ cres 5628 ‘cfv 6494 Basecbs 17174 lecple 17222 LSpanclspn 20961 Atomscatm 39727 HLchlt 39814 LHypclh 40448 LTrncltrn 40565 trLctrl 40622 DVecHcdvh 41542 DIsoHcdih 41692

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-riotaBAD 39417

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7813 df-1st 7937 df-2nd 7938 df-tpos 8171 df-undef 8218 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-er 8638 df-map 8770 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-n0 12433 df-z 12520 df-uz 12784 df-fz 13457 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-sca 17231 df-vsca 17232 df-0g 17399 df-proset 18255 df-poset 18274 df-plt 18289 df-lub 18305 df-glb 18306 df-join 18307 df-meet 18308 df-p0 18384 df-p1 18385 df-lat 18393 df-clat 18460 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18907 df-minusg 18908 df-sbg 18909 df-cmn 19752 df-abl 19753 df-mgp 20117 df-rng 20129 df-ur 20158 df-ring 20211 df-oppr 20312 df-dvdsr 20332 df-unit 20333 df-invr 20363 df-dvr 20376 df-drng 20703 df-lmod 20852 df-lss 20922 df-lsp 20962 df-lvec 21094 df-oposet 39640 df-ol 39642 df-oml 39643 df-covers 39730 df-ats 39731 df-atl 39762 df-cvlat 39786 df-hlat 39815 df-llines 39962 df-lplanes 39963 df-lvols 39964 df-lines 39965 df-psubsp 39967 df-pmap 39968 df-padd 40260 df-lhyp 40452 df-laut 40453 df-ldil 40568 df-ltrn 40569 df-trl 40623 df-tendo 41219 df-edring 41221 df-disoa 41493 df-dvech 41543 df-dib 41603 df-dih 41693

This theorem is referenced by: dihatlat 41798 dihatexv 41802

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