hdmap14lem10 - Mathbox for Norm Megill

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Theorem hdmap14lem10 42684

Description: Part of proof of part 14 in [Baer] p. 49 line 38. (Contributed by NM, 3-Jun-2015.)

**Hypotheses**
Ref	Expression
hdmap14lem8.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmap14lem8.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap14lem8.v	⊢ 𝑉 = (Base‘𝑈)
hdmap14lem8.q	⊢ + = (+_g‘𝑈)
hdmap14lem8.t	⊢ · = ( ·_𝑠 ‘𝑈)
hdmap14lem8.o	⊢ 0 = (0_g‘𝑈)
hdmap14lem8.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmap14lem8.r	⊢ 𝑅 = (Scalar‘𝑈)
hdmap14lem8.b	⊢ 𝐵 = (Base‘𝑅)
hdmap14lem8.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap14lem8.d	⊢ ✚ = (+_g‘𝐶)
hdmap14lem8.e	⊢ ∙ = ( ·_𝑠 ‘𝐶)
hdmap14lem8.p	⊢ 𝑃 = (Scalar‘𝐶)
hdmap14lem8.a	⊢ 𝐴 = (Base‘𝑃)
hdmap14lem8.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmap14lem8.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmap14lem8.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
hdmap14lem8.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
hdmap14lem8.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)
hdmap14lem8.g	⊢ (𝜑 → 𝐺 ∈ 𝐴)
hdmap14lem8.i	⊢ (𝜑 → 𝐼 ∈ 𝐴)
hdmap14lem8.xx	⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)))
hdmap14lem8.yy	⊢ (𝜑 → (𝑆‘(𝐹 · 𝑌)) = (𝐼 ∙ (𝑆‘𝑌)))
hdmap14lem8.ne	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))

**Assertion**
Ref	Expression
hdmap14lem10	⊢ (𝜑 → 𝐺 = 𝐼)

Proof of Theorem hdmap14lem10 Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hdmap14lem8.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmap14lem8.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmap14lem8.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
4		hdmap14lem8.t	. . 3 ⊢ · = ( ·_𝑠 ‘𝑈)
5		hdmap14lem8.r	. . 3 ⊢ 𝑅 = (Scalar‘𝑈)
6		hdmap14lem8.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
7		hdmap14lem8.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
8		hdmap14lem8.e	. . 3 ⊢ ∙ = ( ·_𝑠 ‘𝐶)
9		eqid 2765	. . 3 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶)
10		hdmap14lem8.p	. . 3 ⊢ 𝑃 = (Scalar‘𝐶)
11		hdmap14lem8.a	. . 3 ⊢ 𝐴 = (Base‘𝑃)
12		hdmap14lem8.s	. . 3 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
13		hdmap14lem8.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
14	1, 2, 13	dvhlmod 41917	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
15		hdmap14lem8.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
16	15	eldifad 3918	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
17		hdmap14lem8.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
18	17	eldifad 3918	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
19		hdmap14lem8.q	. . . . 5 ⊢ + = (+_g‘𝑈)
20	3, 19	lmodvacl 21026	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
21	14, 16, 18, 20	syl3anc 1398	. . 3 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉)
22		hdmap14lem8.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
23	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 21, 22	hdmap14lem2a 42674	. 2 ⊢ (𝜑 → ∃𝑔 ∈ 𝐴 (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌))))
24		hdmap14lem8.o	. . . 4 ⊢ 0 = (0_g‘𝑈)
25		hdmap14lem8.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑈)
26		hdmap14lem8.d	. . . 4 ⊢ ✚ = (+_g‘𝐶)
27	13	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
28	15	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝑋 ∈ (𝑉 ∖ { 0 }))
29	17	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝑌 ∈ (𝑉 ∖ { 0 }))
30	22	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝐹 ∈ 𝐵)
31		hdmap14lem8.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐴)
32	31	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝐺 ∈ 𝐴)
33		hdmap14lem8.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝐴)
34	33	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝐼 ∈ 𝐴)
35		hdmap14lem8.xx	. . . . 5 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)))
36	35	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)))
37		hdmap14lem8.yy	. . . . 5 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑌)) = (𝐼 ∙ (𝑆‘𝑌)))
38	37	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → (𝑆‘(𝐹 · 𝑌)) = (𝐼 ∙ (𝑆‘𝑌)))
39		hdmap14lem8.ne	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
40	39	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
41		simp2 1155	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝑔 ∈ 𝐴)
42		simp3 1156	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌))))
43	1, 2, 3, 19, 4, 24, 25, 5, 6, 7, 26, 8, 10, 11, 12, 27, 28, 29, 30, 32, 34, 36, 38, 40, 41, 42	hdmap14lem9 42683	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝐺 = 𝐼)
44	43	rexlimdv3a 3172	. 2 ⊢ (𝜑 → (∃𝑔 ∈ 𝐴 (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌))) → 𝐺 = 𝐼))
45	23, 44	mpd 16	1 ⊢ (𝜑 → 𝐺 = 𝐼)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ∃wrex 3091 ∖ cdif 3903 {csn 4591 ‘cfv 6540 (class class class)co 7416 Basecbs 17287 +_gcplusg 17328 Scalarcsca 17331 ·_𝑠 cvsca 17332 0_gc0g 17510 LModclmod 21011 LSpanclspn 21122 HLchlt 40157 LHypclh 40791 DVecHcdvh 41885 LCDualclcd 42393 HDMapchdma 42599

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188 ax-riotaBAD 39760

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-ot 4600 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7680 df-om 7865 df-1st 7988 df-2nd 7989 df-tpos 8224 df-undef 8271 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-nn 12245 df-2 12314 df-3 12315 df-4 12316 df-5 12317 df-6 12318 df-n0 12516 df-z 12603 df-uz 12875 df-fz 13548 df-struct 17225 df-sets 17242 df-slot 17260 df-ndx 17272 df-base 17288 df-ress 17309 df-plusg 17341 df-mulr 17342 df-sca 17344 df-vsca 17345 df-0g 17512 df-mre 17656 df-mrc 17657 df-acs 17659 df-proset 18368 df-poset 18387 df-plt 18402 df-lub 18418 df-glb 18419 df-join 18420 df-meet 18421 df-p0 18497 df-p1 18498 df-lat 18506 df-clat 18573 df-mgm 18716 df-sgrp 18799 df-mnd 18815 df-submnd 18866 df-grp 19027 df-minusg 19028 df-sbg 19029 df-subg 19213 df-cntz 19411 df-oppg 19440 df-lsm 19730 df-cmn 19876 df-abl 19877 df-mgp 20241 df-rng 20255 df-ur 20288 df-ring 20341 df-oppr 20445 df-dvdsr 20465 df-unit 20466 df-invr 20496 df-dvr 20509 df-nzr 20640 df-rlreg 20823 df-domn 20824 df-drng 20859 df-lmod 21013 df-lss 21083 df-lsp 21123 df-lvec 21254 df-lsatoms 39783 df-lshyp 39784 df-lcv 39826 df-lfl 39865 df-lkr 39893 df-ldual 39931 df-oposet 39983 df-ol 39985 df-oml 39986 df-covers 40073 df-ats 40074 df-atl 40105 df-cvlat 40129 df-hlat 40158 df-llines 40305 df-lplanes 40306 df-lvols 40307 df-lines 40308 df-psubsp 40310 df-pmap 40311 df-padd 40603 df-lhyp 40795 df-laut 40796 df-ldil 40911 df-ltrn 40912 df-trl 40966 df-tgrp 41550 df-tendo 41562 df-edring 41564 df-dveca 41810 df-disoa 41836 df-dvech 41886 df-dib 41946 df-dic 41980 df-dih 42036 df-doch 42155 df-djh 42202 df-lcdual 42394 df-mapd 42432 df-hvmap 42564 df-hdmap1 42600 df-hdmap 42601

This theorem is used by: hdmap14lem11 42685

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