hdmap14lem10 - Mathbox for Norm Megill

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Theorem hdmap14lem10 41854

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 2734	. . 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 41087	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
15		hdmap14lem8.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
16	15	eldifad 3943	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
17		hdmap14lem8.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
18	17	eldifad 3943	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
19		hdmap14lem8.q	. . . . 5 ⊢ + = (+_g‘𝑈)
20	3, 19	lmodvacl 20842	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
21	14, 16, 18, 20	syl3anc 1372	. . 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 41844	. 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 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
28	15	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝑋 ∈ (𝑉 ∖ { 0 }))
29	17	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝑌 ∈ (𝑉 ∖ { 0 }))
30	22	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝐹 ∈ 𝐵)
31		hdmap14lem8.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐴)
32	31	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝐺 ∈ 𝐴)
33		hdmap14lem8.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝐴)
34	33	3ad2ant1 1133	. . . 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 1133	. . . 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 1133	. . . 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 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
41		simp2 1137	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝑔 ∈ 𝐴)
42		simp3 1138	. . . 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 41853	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝐺 = 𝐼)
44	43	rexlimdv3a 3146	. 2 ⊢ (𝜑 → (∃𝑔 ∈ 𝐴 (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌))) → 𝐺 = 𝐼))
45	23, 44	mpd 15	1 ⊢ (𝜑 → 𝐺 = 𝐼)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2931 ∃wrex 3059 ∖ cdif 3928 {csn 4606 ‘cfv 6541 (class class class)co 7413 Basecbs 17230 +_gcplusg 17274 Scalarcsca 17277 ·_𝑠 cvsca 17278 0_gc0g 17456 LModclmod 20827 LSpanclspn 20938 HLchlt 39326 LHypclh 39961 DVecHcdvh 41055 LCDualclcd 41563 HDMapchdma 41769

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-pre-mulgt0 11214 ax-riotaBAD 38929

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-ot 4615 df-uni 4888 df-int 4927 df-iun 4973 df-iin 4974 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-om 7870 df-1st 7996 df-2nd 7997 df-tpos 8233 df-undef 8280 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8727 df-map 8850 df-en 8968 df-dom 8969 df-sdom 8970 df-fin 8971 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-sub 11476 df-neg 11477 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-n0 12510 df-z 12597 df-uz 12861 df-fz 13530 df-struct 17167 df-sets 17184 df-slot 17202 df-ndx 17214 df-base 17231 df-ress 17254 df-plusg 17287 df-mulr 17288 df-sca 17290 df-vsca 17291 df-0g 17458 df-mre 17601 df-mrc 17602 df-acs 17604 df-proset 18311 df-poset 18330 df-plt 18345 df-lub 18361 df-glb 18362 df-join 18363 df-meet 18364 df-p0 18440 df-p1 18441 df-lat 18447 df-clat 18514 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-grp 18924 df-minusg 18925 df-sbg 18926 df-subg 19111 df-cntz 19305 df-oppg 19334 df-lsm 19623 df-cmn 19769 df-abl 19770 df-mgp 20107 df-rng 20119 df-ur 20148 df-ring 20201 df-oppr 20303 df-dvdsr 20326 df-unit 20327 df-invr 20357 df-dvr 20370 df-nzr 20482 df-rlreg 20663 df-domn 20664 df-drng 20700 df-lmod 20829 df-lss 20899 df-lsp 20939 df-lvec 21071 df-lsatoms 38952 df-lshyp 38953 df-lcv 38995 df-lfl 39034 df-lkr 39062 df-ldual 39100 df-oposet 39152 df-ol 39154 df-oml 39155 df-covers 39242 df-ats 39243 df-atl 39274 df-cvlat 39298 df-hlat 39327 df-llines 39475 df-lplanes 39476 df-lvols 39477 df-lines 39478 df-psubsp 39480 df-pmap 39481 df-padd 39773 df-lhyp 39965 df-laut 39966 df-ldil 40081 df-ltrn 40082 df-trl 40136 df-tgrp 40720 df-tendo 40732 df-edring 40734 df-dveca 40980 df-disoa 41006 df-dvech 41056 df-dib 41116 df-dic 41150 df-dih 41206 df-doch 41325 df-djh 41372 df-lcdual 41564 df-mapd 41602 df-hvmap 41734 df-hdmap1 41770 df-hdmap 41771

This theorem is referenced by: hdmap14lem11 41855

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