hdmap14lem12 - Mathbox for Norm Megill

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Theorem hdmap14lem12 40342

Description: Lemma for proof of part 14 in [Baer] p. 50. (Contributed by NM, 6-Jun-2015.)

**Hypotheses**
Ref	Expression
hdmap14lem12.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmap14lem12.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap14lem12.v	⊢ 𝑉 = (Base‘𝑈)
hdmap14lem12.t	⊢ · = ( ·_𝑠 ‘𝑈)
hdmap14lem12.r	⊢ 𝑅 = (Scalar‘𝑈)
hdmap14lem12.b	⊢ 𝐵 = (Base‘𝑅)
hdmap14lem12.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap14lem12.e	⊢ ∙ = ( ·_𝑠 ‘𝐶)
hdmap14lem12.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmap14lem12.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmap14lem12.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)
hdmap14lem12.p	⊢ 𝑃 = (Scalar‘𝐶)
hdmap14lem12.a	⊢ 𝐴 = (Base‘𝑃)
hdmap14lem12.o	⊢ 0 = (0_g‘𝑈)
hdmap14lem12.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
hdmap14lem12.g	⊢ (𝜑 → 𝐺 ∈ 𝐴)

**Assertion**
Ref	Expression
hdmap14lem12	⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ↔ ∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))))

Distinct variable groups: 𝑦,𝐴 𝑦, ∙ 𝑦,𝐹 𝑦,𝐺 𝑦, 0 𝑦,𝑆 𝑦, · 𝑦,𝑈 𝑦,𝑉 𝑦,𝑋 𝜑,𝑦 Allowed substitution hints: 𝐵(𝑦) 𝐶(𝑦) 𝑃(𝑦) 𝑅(𝑦) 𝐻(𝑦) 𝐾(𝑦) 𝑊(𝑦)

Proof of Theorem hdmap14lem12 Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hdmap14lem12.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmap14lem12.u	. . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmap14lem12.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑈)
4		hdmap14lem12.t	. . . . . 6 ⊢ · = ( ·_𝑠 ‘𝑈)
5		hdmap14lem12.r	. . . . . 6 ⊢ 𝑅 = (Scalar‘𝑈)
6		hdmap14lem12.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
7		hdmap14lem12.c	. . . . . 6 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
8		hdmap14lem12.e	. . . . . 6 ⊢ ∙ = ( ·_𝑠 ‘𝐶)
9		eqid 2736	. . . . . 6 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶)
10		hdmap14lem12.p	. . . . . 6 ⊢ 𝑃 = (Scalar‘𝐶)
11		hdmap14lem12.a	. . . . . 6 ⊢ 𝐴 = (Base‘𝑃)
12		hdmap14lem12.s	. . . . . 6 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
13		hdmap14lem12.k	. . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
14	13	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15		simp3 1138	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → 𝑦 ∈ (𝑉 ∖ { 0 }))
16	15	eldifad 3922	. . . . . 6 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → 𝑦 ∈ 𝑉)
17		hdmap14lem12.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
18	17	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → 𝐹 ∈ 𝐵)
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18	hdmap14lem2a 40330	. . . . 5 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → ∃𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦)))
20		simp3 1138	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦)))
21		eqid 2736	. . . . . . . . 9 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
22		hdmap14lem12.o	. . . . . . . . 9 ⊢ 0 = (0_g‘𝑈)
23		eqid 2736	. . . . . . . . 9 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
24		eqid 2736	. . . . . . . . 9 ⊢ (+_g‘𝐶) = (+_g‘𝐶)
25		simp11 1203	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → 𝜑)
26	25, 13	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
27		hdmap14lem12.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
28	25, 27	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → 𝑋 ∈ (𝑉 ∖ { 0 }))
29		simp13 1205	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → 𝑦 ∈ (𝑉 ∖ { 0 }))
30	25, 17	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → 𝐹 ∈ 𝐵)
31		hdmap14lem12.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ 𝐴)
32	25, 31	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → 𝐺 ∈ 𝐴)
33		simp2 1137	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → 𝑔 ∈ 𝐴)
34		simp12 1204	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)))
35	1, 2, 3, 21, 4, 22, 23, 5, 6, 7, 24, 8, 10, 11, 12, 26, 28, 29, 30, 32, 33, 34, 20	hdmap14lem11 40341	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → 𝐺 = 𝑔)
36	35	oveq1d 7372	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → (𝐺 ∙ (𝑆‘𝑦)) = (𝑔 ∙ (𝑆‘𝑦)))
37	20, 36	eqtr4d 2779	. . . . . 6 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))
38	37	rexlimdv3a 3156	. . . . 5 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → (∃𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦)) → (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))))
39	19, 38	mpd 15	. . . 4 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))
40	39	3expia 1121	. . 3 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) → (𝑦 ∈ (𝑉 ∖ { 0 }) → (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))))
41	40	ralrimiv 3142	. 2 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) → ∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))
42		oveq2 7365	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝐹 · 𝑦) = (𝐹 · 𝑋))
43	42	fveq2d 6846	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑆‘(𝐹 · 𝑦)) = (𝑆‘(𝐹 · 𝑋)))
44		fveq2 6842	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝑆‘𝑦) = (𝑆‘𝑋))
45	44	oveq2d 7373	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝐺 ∙ (𝑆‘𝑦)) = (𝐺 ∙ (𝑆‘𝑋)))
46	43, 45	eqeq12d 2752	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) ↔ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))))
47	46	rspcv 3577	. . . 4 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → (∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))))
48	27, 47	syl 17	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))))
49	48	imp 407	. 2 ⊢ ((𝜑 ∧ ∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))) → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)))
50	41, 49	impbida 799	1 ⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ↔ ∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∃wrex 3073 ∖ cdif 3907 {csn 4586 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 +_gcplusg 17133 Scalarcsca 17136 ·_𝑠 cvsca 17137 0_gc0g 17321 LSpanclspn 20432 HLchlt 37812 LHypclh 38447 DVecHcdvh 39541 LCDualclcd 40049 HDMapchdma 40255

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-riotaBAD 37415

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-ot 4595 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-tpos 8157 df-undef 8204 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-0g 17323 df-mre 17466 df-mrc 17467 df-acs 17469 df-proset 18184 df-poset 18202 df-plt 18219 df-lub 18235 df-glb 18236 df-join 18237 df-meet 18238 df-p0 18314 df-p1 18315 df-lat 18321 df-clat 18388 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-subg 18925 df-cntz 19097 df-oppg 19124 df-lsm 19418 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-drng 20187 df-lmod 20324 df-lss 20393 df-lsp 20433 df-lvec 20564 df-lsatoms 37438 df-lshyp 37439 df-lcv 37481 df-lfl 37520 df-lkr 37548 df-ldual 37586 df-oposet 37638 df-ol 37640 df-oml 37641 df-covers 37728 df-ats 37729 df-atl 37760 df-cvlat 37784 df-hlat 37813 df-llines 37961 df-lplanes 37962 df-lvols 37963 df-lines 37964 df-psubsp 37966 df-pmap 37967 df-padd 38259 df-lhyp 38451 df-laut 38452 df-ldil 38567 df-ltrn 38568 df-trl 38622 df-tgrp 39206 df-tendo 39218 df-edring 39220 df-dveca 39466 df-disoa 39492 df-dvech 39542 df-dib 39602 df-dic 39636 df-dih 39692 df-doch 39811 df-djh 39858 df-lcdual 40050 df-mapd 40088 df-hvmap 40220 df-hdmap1 40256 df-hdmap 40257

This theorem is referenced by: hdmap14lem13 40343

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