hdmap14lem10 - Mathbox for Norm Megill

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Theorem hdmap14lem10 35970

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 2609	. . 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 35200	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
15		hdmap14lem8.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
16	15	eldifad 3551	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
17		hdmap14lem8.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
18	17	eldifad 3551	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
19		hdmap14lem8.q	. . . . 5 ⊢ + = (+_g‘𝑈)
20	3, 19	lmodvacl 18648	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
21	14, 16, 18, 20	syl3anc 1317	. . 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 35960	. 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 1074	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
28	15	3ad2ant1 1074	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝑋 ∈ (𝑉 ∖ { 0 }))
29	17	3ad2ant1 1074	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝑌 ∈ (𝑉 ∖ { 0 }))
30	22	3ad2ant1 1074	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝐹 ∈ 𝐵)
31		hdmap14lem8.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐴)
32	31	3ad2ant1 1074	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝐺 ∈ 𝐴)
33		hdmap14lem8.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝐴)
34	33	3ad2ant1 1074	. . . 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 1074	. . . 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 1074	. . . 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 1074	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
41		simp2 1054	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝑔 ∈ 𝐴)
42		simp3 1055	. . . 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 35969	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝐺 = 𝐼)
44	43	rexlimdv3a 3014	. 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 382 ∧ w3a 1030 = wceq 1474 ∈ wcel 1976 ≠ wne 2779 ∃wrex 2896 ∖ cdif 3536 {csn 4124 ‘cfv 5789 (class class class)co 6526 Basecbs 15643 +_gcplusg 15716 Scalarcsca 15719 ·_𝑠 cvsca 15720 0_gc0g 15871 LModclmod 18634 LSpanclspn 18740 HLchlt 33438 LHypclh 34071 DVecHcdvh 35168 LCDualclcd 35676 HDMapchdma 35883

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1712 ax-4 1727 ax-5 1826 ax-6 1874 ax-7 1921 ax-8 1978 ax-9 1985 ax-10 2005 ax-11 2020 ax-12 2033 ax-13 2233 ax-ext 2589 ax-rep 4693 ax-sep 4703 ax-nul 4711 ax-pow 4763 ax-pr 4827 ax-un 6824 ax-cnex 9848 ax-resscn 9849 ax-1cn 9850 ax-icn 9851 ax-addcl 9852 ax-addrcl 9853 ax-mulcl 9854 ax-mulrcl 9855 ax-mulcom 9856 ax-addass 9857 ax-mulass 9858 ax-distr 9859 ax-i2m1 9860 ax-1ne0 9861 ax-1rid 9862 ax-rnegex 9863 ax-rrecex 9864 ax-cnre 9865 ax-pre-lttri 9866 ax-pre-lttrn 9867 ax-pre-ltadd 9868 ax-pre-mulgt0 9869 ax-riotaBAD 33040

This theorem depends on definitions: df-bi 195 df-or 383 df-an 384 df-3or 1031 df-3an 1032 df-tru 1477 df-fal 1480 df-ex 1695 df-nf 1700 df-sb 1867 df-eu 2461 df-mo 2462 df-clab 2596 df-cleq 2602 df-clel 2605 df-nfc 2739 df-ne 2781 df-nel 2782 df-ral 2900 df-rex 2901 df-reu 2902 df-rmo 2903 df-rab 2904 df-v 3174 df-sbc 3402 df-csb 3499 df-dif 3542 df-un 3544 df-in 3546 df-ss 3553 df-pss 3555 df-nul 3874 df-if 4036 df-pw 4109 df-sn 4125 df-pr 4127 df-tp 4129 df-op 4131 df-ot 4133 df-uni 4367 df-int 4405 df-iun 4451 df-iin 4452 df-br 4578 df-opab 4638 df-mpt 4639 df-tr 4675 df-eprel 4938 df-id 4942 df-po 4948 df-so 4949 df-fr 4986 df-we 4988 df-xp 5033 df-rel 5034 df-cnv 5035 df-co 5036 df-dm 5037 df-rn 5038 df-res 5039 df-ima 5040 df-pred 5582 df-ord 5628 df-on 5629 df-lim 5630 df-suc 5631 df-iota 5753 df-fun 5791 df-fn 5792 df-f 5793 df-f1 5794 df-fo 5795 df-f1o 5796 df-fv 5797 df-riota 6488 df-ov 6529 df-oprab 6530 df-mpt2 6531 df-of 6772 df-om 6935 df-1st 7036 df-2nd 7037 df-tpos 7216 df-undef 7263 df-wrecs 7271 df-recs 7332 df-rdg 7370 df-1o 7424 df-oadd 7428 df-er 7606 df-map 7723 df-en 7819 df-dom 7820 df-sdom 7821 df-fin 7822 df-pnf 9932 df-mnf 9933 df-xr 9934 df-ltxr 9935 df-le 9936 df-sub 10119 df-neg 10120 df-nn 10870 df-2 10928 df-3 10929 df-4 10930 df-5 10931 df-6 10932 df-n0 11142 df-z 11213 df-uz 11522 df-fz 12155 df-struct 15645 df-ndx 15646 df-slot 15647 df-base 15648 df-sets 15649 df-ress 15650 df-plusg 15729 df-mulr 15730 df-sca 15732 df-vsca 15733 df-0g 15873 df-mre 16017 df-mrc 16018 df-acs 16020 df-preset 16699 df-poset 16717 df-plt 16729 df-lub 16745 df-glb 16746 df-join 16747 df-meet 16748 df-p0 16810 df-p1 16811 df-lat 16817 df-clat 16879 df-mgm 17013 df-sgrp 17055 df-mnd 17066 df-submnd 17107 df-grp 17196 df-minusg 17197 df-sbg 17198 df-subg 17362 df-cntz 17521 df-oppg 17547 df-lsm 17822 df-cmn 17966 df-abl 17967 df-mgp 18261 df-ur 18273 df-ring 18320 df-oppr 18394 df-dvdsr 18412 df-unit 18413 df-invr 18443 df-dvr 18454 df-drng 18520 df-lmod 18636 df-lss 18702 df-lsp 18741 df-lvec 18872 df-lsatoms 33064 df-lshyp 33065 df-lcv 33107 df-lfl 33146 df-lkr 33174 df-ldual 33212 df-oposet 33264 df-ol 33266 df-oml 33267 df-covers 33354 df-ats 33355 df-atl 33386 df-cvlat 33410 df-hlat 33439 df-llines 33585 df-lplanes 33586 df-lvols 33587 df-lines 33588 df-psubsp 33590 df-pmap 33591 df-padd 33883 df-lhyp 34075 df-laut 34076 df-ldil 34191 df-ltrn 34192 df-trl 34247 df-tgrp 34832 df-tendo 34844 df-edring 34846 df-dveca 35092 df-disoa 35119 df-dvech 35169 df-dib 35229 df-dic 35263 df-dih 35319 df-doch 35438 df-djh 35485 df-lcdual 35677 df-mapd 35715 df-hvmap 35847 df-hdmap1 35884 df-hdmap 35885

This theorem is referenced by: hdmap14lem11 35971

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