dih1dimb2 - Mathbox for Norm Megill

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Theorem dih1dimb2 42441

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 2763	. . 3 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	cdlemf 41763	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄)
7		simp3 1156	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄)
8		simp1rl 1257	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → 𝑄 ∈ 𝐴)
9	7, 8	eqeltrd 2864	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (((trL‘𝐾)‘𝑊)‘𝑓) ∈ 𝐴)
10		simp1l 1216	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11		simp2 1155	. . . . . . 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 41377	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ 𝑇) → (𝑓 ≠ ( I ↾ 𝐵) ↔ (((trL‘𝐾)‘𝑊)‘𝑓) ∈ 𝐴))
14	10, 11, 13	syl2anc 596	. . . . . 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 260	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → 𝑓 ≠ ( I ↾ 𝐵))
16	7	fveq2d 6900	. . . . . 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 42440	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ 𝑇) → (𝐼‘(((trL‘𝐾)‘𝑊)‘𝑓)) = (𝑁‘{⟨𝑓, 𝑂⟩}))
22	10, 11, 21	syl2anc 596	. . . . . 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 2800	. . . . 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 521	. . . 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 1139	. . 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 3179	. 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 16	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2146 ≠ wne 2959 ∃wrex 3090 {csn 4588 ⟨cop 4838 class class class wbr 5106 ↦ cmpt 5189 I cid 5549 ↾ cres 5657 ‘cfv 6547 Basecbs 17402 lecple 17450 LSpanclspn 21261 Atomscatm 40463 HLchlt 40550 LHypclh 41184 LTrncltrn 41301 trLctrl 41358 DVecHcdvh 42278 DIsoHcdih 42428

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7761 ax-cnex 11269 ax-resscn 11270 ax-1cn 11271 ax-icn 11272 ax-addcl 11273 ax-addrcl 11274 ax-mulcl 11275 ax-mulrcl 11276 ax-mulcom 11277 ax-addass 11278 ax-mulass 11279 ax-distr 11280 ax-i2m1 11281 ax-1ne0 11282 ax-1rid 11283 ax-rnegex 11284 ax-rrecex 11285 ax-cnre 11286 ax-pre-lttri 11287 ax-pre-lttrn 11288 ax-pre-ltadd 11289 ax-pre-mulgt0 11290 ax-riotaBAD 40153

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6311 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6501 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7888 df-1st 8011 df-2nd 8012 df-tpos 8257 df-undef 8304 df-frecs 8313 df-wrecs 8344 df-recs 8393 df-rdg 8432 df-1o 8490 df-er 8731 df-map 8863 df-en 8988 df-dom 8989 df-sdom 8990 df-fin 8991 df-pnf 11358 df-mnf 11359 df-xr 11360 df-ltxr 11361 df-le 11362 df-sub 11556 df-neg 11557 df-nn 12349 df-2 12418 df-3 12419 df-4 12420 df-5 12421 df-6 12422 df-n0 12620 df-z 12707 df-uz 12979 df-fz 13653 df-struct 17340 df-sets 17357 df-slot 17375 df-ndx 17387 df-base 17403 df-ress 17424 df-plusg 17456 df-mulr 17457 df-sca 17459 df-vsca 17460 df-0g 17627 df-proset 18483 df-poset 18502 df-plt 18517 df-lub 18533 df-glb 18534 df-join 18535 df-meet 18536 df-p0 18612 df-p1 18613 df-lat 18621 df-clat 18688 df-mgm 18831 df-sgrp 18923 df-mnd 18939 df-grp 19162 df-minusg 19163 df-sbg 19164 df-cmn 20011 df-abl 20012 df-mgp 20376 df-rng 20390 df-ur 20423 df-ring 20476 df-oppr 20582 df-dvdsr 20602 df-unit 20603 df-invr 20633 df-dvr 20646 df-drng 20997 df-lmod 21152 df-lss 21222 df-lsp 21262 df-lvec 21393 df-oposet 40376 df-ol 40378 df-oml 40379 df-covers 40466 df-ats 40467 df-atl 40498 df-cvlat 40522 df-hlat 40551 df-llines 40698 df-lplanes 40699 df-lvols 40700 df-lines 40701 df-psubsp 40703 df-pmap 40704 df-padd 40996 df-lhyp 41188 df-laut 41189 df-ldil 41304 df-ltrn 41305 df-trl 41359 df-tendo 41955 df-edring 41957 df-disoa 42229 df-dvech 42279 df-dib 42339 df-dih 42429

This theorem is used by: dihatlat 42534 dihatexv 42538

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