dia2dimlem7 - Mathbox for Norm Megill

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Theorem dia2dimlem7 41450

Description: Lemma for dia2dim 41457. Eliminate (𝐹‘𝑃) ≠ 𝑃 condition. (Contributed by NM, 8-Sep-2014.)

**Hypotheses**
Ref	Expression
dia2dimlem7.l	⊢ ≤ = (le‘𝐾)
dia2dimlem7.j	⊢ ∨ = (join‘𝐾)
dia2dimlem7.m	⊢ ∧ = (meet‘𝐾)
dia2dimlem7.a	⊢ 𝐴 = (Atoms‘𝐾)
dia2dimlem7.h	⊢ 𝐻 = (LHyp‘𝐾)
dia2dimlem7.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dia2dimlem7.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
dia2dimlem7.y	⊢ 𝑌 = ((DVecA‘𝐾)‘𝑊)
dia2dimlem7.s	⊢ 𝑆 = (LSubSp‘𝑌)
dia2dimlem7.pl	⊢ ⊕ = (LSSum‘𝑌)
dia2dimlem7.n	⊢ 𝑁 = (LSpan‘𝑌)
dia2dimlem7.i	⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
dia2dimlem7.q	⊢ 𝑄 = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))
dia2dimlem7.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dia2dimlem7.u	⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
dia2dimlem7.v	⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
dia2dimlem7.p	⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
dia2dimlem7.f	⊢ (𝜑 → 𝐹 ∈ 𝑇)
dia2dimlem7.rf	⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))
dia2dimlem7.uv	⊢ (𝜑 → 𝑈 ≠ 𝑉)
dia2dimlem7.ru	⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈)
dia2dimlem7.rv	⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑉)

**Assertion**
Ref	Expression
dia2dimlem7	⊢ (𝜑 → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))

**Proof of Theorem dia2dimlem7**
Step	Hyp	Ref	Expression
1		dia2dimlem7.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		dia2dimlem7.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝑇)
3		dia2dimlem7.p	. . . . 5 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
4		eqid 2737	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		dia2dimlem7.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
6		dia2dimlem7.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
7		dia2dimlem7.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
8		dia2dimlem7.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
9	4, 5, 6, 7, 8	ltrnideq 40555	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹 = ( I ↾ (Base‘𝐾)) ↔ (𝐹‘𝑃) = 𝑃))
10	1, 2, 3, 9	syl3anc 1374	. . . 4 ⊢ (𝜑 → (𝐹 = ( I ↾ (Base‘𝐾)) ↔ (𝐹‘𝑃) = 𝑃))
11		dia2dimlem7.y	. . . . . . . 8 ⊢ 𝑌 = ((DVecA‘𝐾)‘𝑊)
12		eqid 2737	. . . . . . . 8 ⊢ (0_g‘𝑌) = (0_g‘𝑌)
13	4, 7, 8, 11, 12	dva0g 41407	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (0_g‘𝑌) = ( I ↾ (Base‘𝐾)))
14	1, 13	syl 17	. . . . . 6 ⊢ (𝜑 → (0_g‘𝑌) = ( I ↾ (Base‘𝐾)))
15	7, 11	dvalvec 41406	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑌 ∈ LVec)
16		lveclmod 21073	. . . . . . . 8 ⊢ (𝑌 ∈ LVec → 𝑌 ∈ LMod)
17	1, 15, 16	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ LMod)
18		dia2dimlem7.u	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
19	18	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
20	4, 6	atbase 39669	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
21	19, 20	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
22	18	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ≤ 𝑊)
23		dia2dimlem7.i	. . . . . . . . . 10 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
24		dia2dimlem7.s	. . . . . . . . . 10 ⊢ 𝑆 = (LSubSp‘𝑌)
25	4, 5, 7, 11, 23, 24	dialss 41426	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ (Base‘𝐾) ∧ 𝑈 ≤ 𝑊)) → (𝐼‘𝑈) ∈ 𝑆)
26	1, 21, 22, 25	syl12anc 837	. . . . . . . 8 ⊢ (𝜑 → (𝐼‘𝑈) ∈ 𝑆)
27		dia2dimlem7.v	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
28	27	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
29	4, 6	atbase 39669	. . . . . . . . . 10 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
30	28, 29	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾))
31	27	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ≤ 𝑊)
32	4, 5, 7, 11, 23, 24	dialss 41426	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑉 ∈ (Base‘𝐾) ∧ 𝑉 ≤ 𝑊)) → (𝐼‘𝑉) ∈ 𝑆)
33	1, 30, 31, 32	syl12anc 837	. . . . . . . 8 ⊢ (𝜑 → (𝐼‘𝑉) ∈ 𝑆)
34		dia2dimlem7.pl	. . . . . . . . 9 ⊢ ⊕ = (LSSum‘𝑌)
35	24, 34	lsmcl 21050	. . . . . . . 8 ⊢ ((𝑌 ∈ LMod ∧ (𝐼‘𝑈) ∈ 𝑆 ∧ (𝐼‘𝑉) ∈ 𝑆) → ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)) ∈ 𝑆)
36	17, 26, 33, 35	syl3anc 1374	. . . . . . 7 ⊢ (𝜑 → ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)) ∈ 𝑆)
37	12, 24	lss0cl 20913	. . . . . . 7 ⊢ ((𝑌 ∈ LMod ∧ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)) ∈ 𝑆) → (0_g‘𝑌) ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
38	17, 36, 37	syl2anc 585	. . . . . 6 ⊢ (𝜑 → (0_g‘𝑌) ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
39	14, 38	eqeltrrd 2838	. . . . 5 ⊢ (𝜑 → ( I ↾ (Base‘𝐾)) ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
40		eleq1a 2832	. . . . 5 ⊢ (( I ↾ (Base‘𝐾)) ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)) → (𝐹 = ( I ↾ (Base‘𝐾)) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉))))
41	39, 40	syl 17	. . . 4 ⊢ (𝜑 → (𝐹 = ( I ↾ (Base‘𝐾)) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉))))
42	10, 41	sylbird 260	. . 3 ⊢ (𝜑 → ((𝐹‘𝑃) = 𝑃 → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉))))
43	42	imp 406	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝑃) = 𝑃) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
44		dia2dimlem7.j	. . 3 ⊢ ∨ = (join‘𝐾)
45		dia2dimlem7.m	. . 3 ⊢ ∧ = (meet‘𝐾)
46		dia2dimlem7.r	. . 3 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
47		dia2dimlem7.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑌)
48		dia2dimlem7.q	. . 3 ⊢ 𝑄 = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))
49	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
50	18	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
51	27	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
52	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
53	2	anim1i 616	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃))
54		dia2dimlem7.rf	. . . 4 ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))
55	54	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))
56		dia2dimlem7.uv	. . . 4 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
57	56	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑃) ≠ 𝑃) → 𝑈 ≠ 𝑉)
58		dia2dimlem7.ru	. . . 4 ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈)
59	58	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝑅‘𝐹) ≠ 𝑈)
60		dia2dimlem7.rv	. . . 4 ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑉)
61	60	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝑅‘𝐹) ≠ 𝑉)
62	5, 44, 45, 6, 7, 8, 46, 11, 24, 34, 47, 23, 48, 49, 50, 51, 52, 53, 55, 57, 59, 61	dia2dimlem6 41449	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝑃) ≠ 𝑃) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
63	43, 62	pm2.61dane 3020	1 ⊢ (𝜑 → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 class class class wbr 5100 I cid 5526 ↾ cres 5634 ‘cfv 6500 (class class class)co 7368 Basecbs 17148 lecple 17196 0_gc0g 17371 joincjn 18246 meetcmee 18247 LSSumclsm 19578 LModclmod 20826 LSubSpclss 20897 LSpanclspn 20937 LVecclvec 21069 Atomscatm 39643 HLchlt 39730 LHypclh 40364 LTrncltrn 40481 trLctrl 40538 DVecAcdveca 41382 DIsoAcdia 41408

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 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-riotaBAD 39333

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 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-tpos 8178 df-undef 8225 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-n0 12414 df-z 12501 df-uz 12764 df-fz 13436 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-sca 17205 df-vsca 17206 df-0g 17373 df-proset 18229 df-poset 18248 df-plt 18263 df-lub 18279 df-glb 18280 df-join 18281 df-meet 18282 df-p0 18358 df-p1 18359 df-lat 18367 df-clat 18434 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-grp 18881 df-minusg 18882 df-sbg 18883 df-subg 19068 df-cntz 19261 df-lsm 19580 df-cmn 19726 df-abl 19727 df-mgp 20091 df-rng 20103 df-ur 20132 df-ring 20185 df-oppr 20288 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-dvr 20352 df-drng 20679 df-lmod 20828 df-lss 20898 df-lsp 20938 df-lvec 21070 df-oposet 39556 df-ol 39558 df-oml 39559 df-covers 39646 df-ats 39647 df-atl 39678 df-cvlat 39702 df-hlat 39731 df-llines 39878 df-lplanes 39879 df-lvols 39880 df-lines 39881 df-psubsp 39883 df-pmap 39884 df-padd 40176 df-lhyp 40368 df-laut 40369 df-ldil 40484 df-ltrn 40485 df-trl 40539 df-tgrp 41123 df-tendo 41135 df-edring 41137 df-dveca 41383 df-disoa 41409

This theorem is referenced by: dia2dimlem8 41451

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