mapdpglem10 - Mathbox for Norm Megill

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Theorem mapdpglem10 41646

Description: Lemma for mapdpg 41671. 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 41074	. 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 41645	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌}))
32	1, 2, 3, 7, 8, 31, 29	lspsneleq 21074	1 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 {csn 4601 ‘cfv 6530 (class class class)co 7403 Basecbs 17226 Scalarcsca 17272 ·_𝑠 cvsca 17273 0_gc0g 17451 -_gcsg 18916 LSSumclsm 19613 LSpanclspn 20926 HLchlt 39314 LHypclh 39949 DVecHcdvh 41043 LCDualclcd 41551 mapdcmpd 41589

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-riotaBAD 38917

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7669 df-om 7860 df-1st 7986 df-2nd 7987 df-tpos 8223 df-undef 8270 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-er 8717 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-n0 12500 df-z 12587 df-uz 12851 df-fz 13523 df-struct 17164 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-ress 17250 df-plusg 17282 df-mulr 17283 df-sca 17285 df-vsca 17286 df-0g 17453 df-mre 17596 df-mrc 17597 df-acs 17599 df-proset 18304 df-poset 18323 df-plt 18338 df-lub 18354 df-glb 18355 df-join 18356 df-meet 18357 df-p0 18433 df-p1 18434 df-lat 18440 df-clat 18507 df-mgm 18616 df-sgrp 18695 df-mnd 18711 df-submnd 18760 df-grp 18917 df-minusg 18918 df-sbg 18919 df-subg 19104 df-cntz 19298 df-oppg 19327 df-lsm 19615 df-cmn 19761 df-abl 19762 df-mgp 20099 df-rng 20111 df-ur 20140 df-ring 20193 df-oppr 20295 df-dvdsr 20315 df-unit 20316 df-invr 20346 df-dvr 20359 df-nzr 20471 df-rlreg 20652 df-domn 20653 df-drng 20689 df-lmod 20817 df-lss 20887 df-lsp 20927 df-lvec 21059 df-lsatoms 38940 df-lshyp 38941 df-lcv 38983 df-lfl 39022 df-lkr 39050 df-ldual 39088 df-oposet 39140 df-ol 39142 df-oml 39143 df-covers 39230 df-ats 39231 df-atl 39262 df-cvlat 39286 df-hlat 39315 df-llines 39463 df-lplanes 39464 df-lvols 39465 df-lines 39466 df-psubsp 39468 df-pmap 39469 df-padd 39761 df-lhyp 39953 df-laut 39954 df-ldil 40069 df-ltrn 40070 df-trl 40124 df-tgrp 40708 df-tendo 40720 df-edring 40722 df-dveca 40968 df-disoa 40994 df-dvech 41044 df-dib 41104 df-dic 41138 df-dih 41194 df-doch 41313 df-djh 41360 df-lcdual 41552 df-mapd 41590

This theorem is referenced by: mapdpglem11 41647

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