hdmap14lem10 - Mathbox for Norm Megill

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Theorem hdmap14lem10 42632

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 2763	. . 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 41865	. . . 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 20977	. . . 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 42622	. 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 42631	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝐺 = 𝐼)
44	43	rexlimdv3a 3170	. 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

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∃wrex 3089 ∖ cdif 3903 {csn 4590 ‘cfv 6538 (class class class)co 7412 Basecbs 17270 +_gcplusg 17311 Scalarcsca 17314 ·_𝑠 cvsca 17315 0_gc0g 17493 LModclmod 20962 LSpanclspn 21073 HLchlt 40105 LHypclh 40739 DVecHcdvh 41833 LCDualclcd 42341 HDMapchdma 42547

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-riotaBAD 39708

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-ot 4599 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 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 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7864 df-1st 7987 df-2nd 7988 df-tpos 8223 df-undef 8270 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-2o 8455 df-er 8695 df-map 8827 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-5 12307 df-6 12308 df-n0 12506 df-z 12593 df-uz 12864 df-fz 13537 df-struct 17208 df-sets 17225 df-slot 17243 df-ndx 17255 df-base 17271 df-ress 17292 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-0g 17495 df-mre 17639 df-mrc 17640 df-acs 17642 df-proset 18351 df-poset 18370 df-plt 18385 df-lub 18401 df-glb 18402 df-join 18403 df-meet 18404 df-p0 18480 df-p1 18481 df-lat 18489 df-clat 18556 df-mgm 18699 df-sgrp 18778 df-mnd 18794 df-submnd 18843 df-grp 19004 df-minusg 19005 df-sbg 19006 df-subg 19190 df-cntz 19388 df-oppg 19417 df-lsm 19707 df-cmn 19853 df-abl 19854 df-mgp 20218 df-rng 20232 df-ur 20265 df-ring 20318 df-oppr 20420 df-dvdsr 20440 df-unit 20441 df-invr 20471 df-dvr 20484 df-nzr 20597 df-rlreg 20780 df-domn 20781 df-drng 20816 df-lmod 20964 df-lss 21034 df-lsp 21074 df-lvec 21205 df-lsatoms 39731 df-lshyp 39732 df-lcv 39774 df-lfl 39813 df-lkr 39841 df-ldual 39879 df-oposet 39931 df-ol 39933 df-oml 39934 df-covers 40021 df-ats 40022 df-atl 40053 df-cvlat 40077 df-hlat 40106 df-llines 40253 df-lplanes 40254 df-lvols 40255 df-lines 40256 df-psubsp 40258 df-pmap 40259 df-padd 40551 df-lhyp 40743 df-laut 40744 df-ldil 40859 df-ltrn 40860 df-trl 40914 df-tgrp 41498 df-tendo 41510 df-edring 41512 df-dveca 41758 df-disoa 41784 df-dvech 41834 df-dib 41894 df-dic 41928 df-dih 41984 df-doch 42103 df-djh 42150 df-lcdual 42342 df-mapd 42380 df-hvmap 42512 df-hdmap1 42548 df-hdmap 42549

This theorem is referenced by: hdmap14lem11 42633

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