dia2dimlem9 - Mathbox for Norm Megill

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Theorem dia2dimlem9 41055

Description: Lemma for dia2dim 41060. Eliminate (𝑅‘𝐹) ≠ 𝑈, 𝑉 conditions. (Contributed by NM, 8-Sep-2014.)

**Hypotheses**
Ref	Expression
dia2dimlem9.l	⊢ ≤ = (le‘𝐾)
dia2dimlem9.j	⊢ ∨ = (join‘𝐾)
dia2dimlem9.m	⊢ ∧ = (meet‘𝐾)
dia2dimlem9.a	⊢ 𝐴 = (Atoms‘𝐾)
dia2dimlem9.h	⊢ 𝐻 = (LHyp‘𝐾)
dia2dimlem9.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dia2dimlem9.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
dia2dimlem9.y	⊢ 𝑌 = ((DVecA‘𝐾)‘𝑊)
dia2dimlem9.s	⊢ 𝑆 = (LSubSp‘𝑌)
dia2dimlem9.pl	⊢ ⊕ = (LSSum‘𝑌)
dia2dimlem9.n	⊢ 𝑁 = (LSpan‘𝑌)
dia2dimlem9.i	⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
dia2dimlem9.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dia2dimlem9.u	⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
dia2dimlem9.v	⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
dia2dimlem9.f	⊢ (𝜑 → 𝐹 ∈ 𝑇)
dia2dimlem9.rf	⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))
dia2dimlem9.uv	⊢ (𝜑 → 𝑈 ≠ 𝑉)

**Assertion**
Ref	Expression
dia2dimlem9	⊢ (𝜑 → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))

**Proof of Theorem dia2dimlem9**
Step	Hyp	Ref	Expression
1		dia2dimlem9.k	. . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		dia2dimlem9.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
3		dia2dimlem9.y	. . . . . . . 8 ⊢ 𝑌 = ((DVecA‘𝐾)‘𝑊)
4	2, 3	dvalvec 41009	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑌 ∈ LVec)
5		lveclmod 21123	. . . . . . 7 ⊢ (𝑌 ∈ LVec → 𝑌 ∈ LMod)
6		dia2dimlem9.s	. . . . . . . 8 ⊢ 𝑆 = (LSubSp‘𝑌)
7	6	lsssssubg 20974	. . . . . . 7 ⊢ (𝑌 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑌))
8	1, 4, 5, 7	4syl 19	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑌))
9		dia2dimlem9.u	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
10	9	simpld 494	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
11		eqid 2735	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		dia2dimlem9.a	. . . . . . . . 9 ⊢ 𝐴 = (Atoms‘𝐾)
13	11, 12	atbase 39271	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
14	10, 13	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
15	9	simprd 495	. . . . . . 7 ⊢ (𝜑 → 𝑈 ≤ 𝑊)
16		dia2dimlem9.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
17		dia2dimlem9.i	. . . . . . . 8 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
18	11, 16, 2, 3, 17, 6	dialss 41029	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ (Base‘𝐾) ∧ 𝑈 ≤ 𝑊)) → (𝐼‘𝑈) ∈ 𝑆)
19	1, 14, 15, 18	syl12anc 837	. . . . . 6 ⊢ (𝜑 → (𝐼‘𝑈) ∈ 𝑆)
20	8, 19	sseldd 3996	. . . . 5 ⊢ (𝜑 → (𝐼‘𝑈) ∈ (SubGrp‘𝑌))
21		dia2dimlem9.v	. . . . . . . . 9 ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
22	21	simpld 494	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
23	11, 12	atbase 39271	. . . . . . . 8 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
24	22, 23	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾))
25	21	simprd 495	. . . . . . 7 ⊢ (𝜑 → 𝑉 ≤ 𝑊)
26	11, 16, 2, 3, 17, 6	dialss 41029	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑉 ∈ (Base‘𝐾) ∧ 𝑉 ≤ 𝑊)) → (𝐼‘𝑉) ∈ 𝑆)
27	1, 24, 25, 26	syl12anc 837	. . . . . 6 ⊢ (𝜑 → (𝐼‘𝑉) ∈ 𝑆)
28	8, 27	sseldd 3996	. . . . 5 ⊢ (𝜑 → (𝐼‘𝑉) ∈ (SubGrp‘𝑌))
29		dia2dimlem9.pl	. . . . . 6 ⊢ ⊕ = (LSSum‘𝑌)
30	29	lsmub1 19690	. . . . 5 ⊢ (((𝐼‘𝑈) ∈ (SubGrp‘𝑌) ∧ (𝐼‘𝑉) ∈ (SubGrp‘𝑌)) → (𝐼‘𝑈) ⊆ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
31	20, 28, 30	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐼‘𝑈) ⊆ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
32	31	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑈) → (𝐼‘𝑈) ⊆ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
33		dia2dimlem9.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑇)
34		dia2dimlem9.t	. . . . . . 7 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
35		dia2dimlem9.r	. . . . . . 7 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
36	2, 34, 35, 17	dia1dimid 41046	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ (𝐼‘(𝑅‘𝐹)))
37	1, 33, 36	syl2anc 584	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐼‘(𝑅‘𝐹)))
38	37	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑈) → 𝐹 ∈ (𝐼‘(𝑅‘𝐹)))
39		fveq2 6907	. . . . 5 ⊢ ((𝑅‘𝐹) = 𝑈 → (𝐼‘(𝑅‘𝐹)) = (𝐼‘𝑈))
40	39	adantl 481	. . . 4 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑈) → (𝐼‘(𝑅‘𝐹)) = (𝐼‘𝑈))
41	38, 40	eleqtrd 2841	. . 3 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑈) → 𝐹 ∈ (𝐼‘𝑈))
42	32, 41	sseldd 3996	. 2 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑈) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
43	20	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑉) → (𝐼‘𝑈) ∈ (SubGrp‘𝑌))
44	28	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑉) → (𝐼‘𝑉) ∈ (SubGrp‘𝑌))
45	29	lsmub2 19691	. . . 4 ⊢ (((𝐼‘𝑈) ∈ (SubGrp‘𝑌) ∧ (𝐼‘𝑉) ∈ (SubGrp‘𝑌)) → (𝐼‘𝑉) ⊆ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
46	43, 44, 45	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑉) → (𝐼‘𝑉) ⊆ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
47	37	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑉) → 𝐹 ∈ (𝐼‘(𝑅‘𝐹)))
48		fveq2 6907	. . . . 5 ⊢ ((𝑅‘𝐹) = 𝑉 → (𝐼‘(𝑅‘𝐹)) = (𝐼‘𝑉))
49	48	adantl 481	. . . 4 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑉) → (𝐼‘(𝑅‘𝐹)) = (𝐼‘𝑉))
50	47, 49	eleqtrd 2841	. . 3 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑉) → 𝐹 ∈ (𝐼‘𝑉))
51	46, 50	sseldd 3996	. 2 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑉) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
52		dia2dimlem9.j	. . 3 ⊢ ∨ = (join‘𝐾)
53		dia2dimlem9.m	. . 3 ⊢ ∧ = (meet‘𝐾)
54		dia2dimlem9.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑌)
55	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
56	9	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
57	21	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
58	33	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → 𝐹 ∈ 𝑇)
59		dia2dimlem9.rf	. . . 4 ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))
60	59	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))
61		dia2dimlem9.uv	. . . 4 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
62	61	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → 𝑈 ≠ 𝑉)
63		simprl 771	. . 3 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → (𝑅‘𝐹) ≠ 𝑈)
64		simprr 773	. . 3 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → (𝑅‘𝐹) ≠ 𝑉)
65	16, 52, 53, 12, 2, 34, 35, 3, 6, 29, 54, 17, 55, 56, 57, 58, 60, 62, 63, 64	dia2dimlem8 41054	. 2 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
66	42, 51, 65	pm2.61da2ne 3028	1 ⊢ (𝜑 → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ⊆ wss 3963 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 lecple 17305 joincjn 18369 meetcmee 18370 SubGrpcsubg 19151 LSSumclsm 19667 LModclmod 20875 LSubSpclss 20947 LSpanclspn 20987 LVecclvec 21119 Atomscatm 39245 HLchlt 39332 LHypclh 39967 LTrncltrn 40084 trLctrl 40141 DVecAcdveca 40985 DIsoAcdia 41011

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-riotaBAD 38935

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-undef 8297 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-0g 17488 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18490 df-clat 18557 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cntz 19348 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-llines 39481 df-lplanes 39482 df-lvols 39483 df-lines 39484 df-psubsp 39486 df-pmap 39487 df-padd 39779 df-lhyp 39971 df-laut 39972 df-ldil 40087 df-ltrn 40088 df-trl 40142 df-tgrp 40726 df-tendo 40738 df-edring 40740 df-dveca 40986 df-disoa 41012

This theorem is referenced by: dia2dimlem11 41057

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