mapdpglem10 - Mathbox for Norm Megill

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Theorem mapdpglem10 39674

Description: Lemma for mapdpg 39699. Baer p. 45, line 6: "Hence Fx=Fy, an impossibility." (Contributed by NM, 20-Mar-2015.)

**Hypotheses**
Ref	Expression
mapdpglem.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdpglem.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdpglem.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdpglem.v	⊢ 𝑉 = (Base‘𝑈)
mapdpglem.s	⊢ − = (-_g‘𝑈)
mapdpglem.n	⊢ 𝑁 = (LSpan‘𝑈)
mapdpglem.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
mapdpglem.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
mapdpglem.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
mapdpglem.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
mapdpglem1.p	⊢ ⊕ = (LSSum‘𝐶)
mapdpglem2.j	⊢ 𝐽 = (LSpan‘𝐶)
mapdpglem3.f	⊢ 𝐹 = (Base‘𝐶)
mapdpglem3.te	⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))))
mapdpglem3.a	⊢ 𝐴 = (Scalar‘𝑈)
mapdpglem3.b	⊢ 𝐵 = (Base‘𝐴)
mapdpglem3.t	⊢ · = ( ·_𝑠 ‘𝐶)
mapdpglem3.r	⊢ 𝑅 = (-_g‘𝐶)
mapdpglem3.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)
mapdpglem3.e	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
mapdpglem4.q	⊢ 𝑄 = (0_g‘𝑈)
mapdpglem.ne	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
mapdpglem4.jt	⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))
mapdpglem4.z	⊢ 0 = (0_g‘𝐴)
mapdpglem4.g4	⊢ (𝜑 → 𝑔 ∈ 𝐵)
mapdpglem4.z4	⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))
mapdpglem4.t4	⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧))
mapdpglem4.xn	⊢ (𝜑 → 𝑋 ≠ 𝑄)
mapdpglem4.g0	⊢ (𝜑 → 𝑔 = 0 )

**Assertion**
Ref	Expression
mapdpglem10	⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))

Distinct variable groups: 𝑡, − 𝑡,𝐶 𝑡,𝐽 𝑡,𝑀 𝑡,𝑁 𝑡,𝑋 𝑡,𝑌 𝐵,𝑔 𝑧,𝑔,𝐶 𝑔,𝐹 𝑔,𝐺,𝑧 𝑔,𝐽,𝑧 𝑔,𝑀,𝑧 𝑔,𝑁,𝑧 𝑅,𝑔,𝑧 · ,𝑔,𝑧 𝑔,𝑌,𝑧,𝑡 Allowed substitution hints: 𝜑(𝑧,𝑡,𝑔) 𝐴(𝑧,𝑡,𝑔) 𝐵(𝑧,𝑡) ⊕ (𝑧,𝑡,𝑔) 𝑄(𝑧,𝑡,𝑔) 𝑅(𝑡) · (𝑡) 𝑈(𝑧,𝑡,𝑔) 𝐹(𝑧,𝑡) 𝐺(𝑡) 𝐻(𝑧,𝑡,𝑔) 𝐾(𝑧,𝑡,𝑔) − (𝑧,𝑔) 𝑉(𝑧,𝑡,𝑔) 𝑊(𝑧,𝑡,𝑔) 𝑋(𝑧,𝑔) 0 (𝑧,𝑡,𝑔)

**Proof of Theorem mapdpglem10**
Step	Hyp	Ref	Expression
1		mapdpglem.v	. 2 ⊢ 𝑉 = (Base‘𝑈)
2		mapdpglem4.q	. 2 ⊢ 𝑄 = (0_g‘𝑈)
3		mapdpglem.n	. 2 ⊢ 𝑁 = (LSpan‘𝑈)
4		mapdpglem.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
5		mapdpglem.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
6		mapdpglem.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
7	4, 5, 6	dvhlvec 39102	. 2 ⊢ (𝜑 → 𝑈 ∈ LVec)
8		mapdpglem.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
9		mapdpglem.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
10		mapdpglem.s	. . 3 ⊢ − = (-_g‘𝑈)
11		mapdpglem.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
12		mapdpglem.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
13		mapdpglem1.p	. . 3 ⊢ ⊕ = (LSSum‘𝐶)
14		mapdpglem2.j	. . 3 ⊢ 𝐽 = (LSpan‘𝐶)
15		mapdpglem3.f	. . 3 ⊢ 𝐹 = (Base‘𝐶)
16		mapdpglem3.te	. . 3 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))))
17		mapdpglem3.a	. . 3 ⊢ 𝐴 = (Scalar‘𝑈)
18		mapdpglem3.b	. . 3 ⊢ 𝐵 = (Base‘𝐴)
19		mapdpglem3.t	. . 3 ⊢ · = ( ·_𝑠 ‘𝐶)
20		mapdpglem3.r	. . 3 ⊢ 𝑅 = (-_g‘𝐶)
21		mapdpglem3.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
22		mapdpglem3.e	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
23		mapdpglem.ne	. . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
24		mapdpglem4.jt	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))
25		mapdpglem4.z	. . 3 ⊢ 0 = (0_g‘𝐴)
26		mapdpglem4.g4	. . 3 ⊢ (𝜑 → 𝑔 ∈ 𝐵)
27		mapdpglem4.z4	. . 3 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))
28		mapdpglem4.t4	. . 3 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧))
29		mapdpglem4.xn	. . 3 ⊢ (𝜑 → 𝑋 ≠ 𝑄)
30		mapdpglem4.g0	. . 3 ⊢ (𝜑 → 𝑔 = 0 )
31	4, 9, 5, 1, 10, 3, 11, 6, 12, 8, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 2, 23, 24, 25, 26, 27, 28, 29, 30	mapdpglem9 39673	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌}))
32	1, 2, 3, 7, 8, 31, 29	lspsneleq 20358	1 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2109 ≠ wne 2944 {csn 4566 ‘cfv 6430 (class class class)co 7268 Basecbs 16893 Scalarcsca 16946 ·_𝑠 cvsca 16947 0_gc0g 17131 -_gcsg 18560 LSSumclsm 19220 LSpanclspn 20214 HLchlt 37343 LHypclh 37977 DVecHcdvh 39071 LCDualclcd 39579 mapdcmpd 39617

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-riotaBAD 36946

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-iin 4932 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-om 7701 df-1st 7817 df-2nd 7818 df-tpos 8026 df-undef 8073 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-map 8591 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-n0 12217 df-z 12303 df-uz 12565 df-fz 13222 df-struct 16829 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-sca 16959 df-vsca 16960 df-0g 17133 df-mre 17276 df-mrc 17277 df-acs 17279 df-proset 17994 df-poset 18012 df-plt 18029 df-lub 18045 df-glb 18046 df-join 18047 df-meet 18048 df-p0 18124 df-p1 18125 df-lat 18131 df-clat 18198 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-submnd 18412 df-grp 18561 df-minusg 18562 df-sbg 18563 df-subg 18733 df-cntz 18904 df-oppg 18931 df-lsm 19222 df-cmn 19369 df-abl 19370 df-mgp 19702 df-ur 19719 df-ring 19766 df-oppr 19843 df-dvdsr 19864 df-unit 19865 df-invr 19895 df-dvr 19906 df-drng 19974 df-lmod 20106 df-lss 20175 df-lsp 20215 df-lvec 20346 df-lsatoms 36969 df-lshyp 36970 df-lcv 37012 df-lfl 37051 df-lkr 37079 df-ldual 37117 df-oposet 37169 df-ol 37171 df-oml 37172 df-covers 37259 df-ats 37260 df-atl 37291 df-cvlat 37315 df-hlat 37344 df-llines 37491 df-lplanes 37492 df-lvols 37493 df-lines 37494 df-psubsp 37496 df-pmap 37497 df-padd 37789 df-lhyp 37981 df-laut 37982 df-ldil 38097 df-ltrn 38098 df-trl 38152 df-tgrp 38736 df-tendo 38748 df-edring 38750 df-dveca 38996 df-disoa 39022 df-dvech 39072 df-dib 39132 df-dic 39166 df-dih 39222 df-doch 39341 df-djh 39388 df-lcdual 39580 df-mapd 39618

This theorem is referenced by: mapdpglem11 39675

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