hdmap14lem12 - Mathbox for Norm Megill

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Theorem hdmap14lem12 41346

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 2728	. . . . . 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 1131	. . . . . 6 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15		simp3 1136	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → 𝑦 ∈ (𝑉 ∖ { 0 }))
16	15	eldifad 3957	. . . . . 6 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → 𝑦 ∈ 𝑉)
17		hdmap14lem12.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
18	17	3ad2ant1 1131	. . . . . 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 41334	. . . . 5 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → ∃𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦)))
20		simp3 1136	. . . . . . 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 2728	. . . . . . . . 9 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
22		hdmap14lem12.o	. . . . . . . . 9 ⊢ 0 = (0_g‘𝑈)
23		eqid 2728	. . . . . . . . 9 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
24		eqid 2728	. . . . . . . . 9 ⊢ (+_g‘𝐶) = (+_g‘𝐶)
25		simp11 1201	. . . . . . . . . 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 1203	. . . . . . . . 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 1135	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → 𝑔 ∈ 𝐴)
34		simp12 1202	. . . . . . . . 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 41345	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → 𝐺 = 𝑔)
36	35	oveq1d 7429	. . . . . . 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 2771	. . . . . 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 3155	. . . . 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 1119	. . 3 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) → (𝑦 ∈ (𝑉 ∖ { 0 }) → (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))))
41	40	ralrimiv 3141	. 2 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) → ∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))
42		oveq2 7422	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝐹 · 𝑦) = (𝐹 · 𝑋))
43	42	fveq2d 6895	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑆‘(𝐹 · 𝑦)) = (𝑆‘(𝐹 · 𝑋)))
44		fveq2 6891	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝑆‘𝑦) = (𝑆‘𝑋))
45	44	oveq2d 7430	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝐺 ∙ (𝑆‘𝑦)) = (𝐺 ∙ (𝑆‘𝑋)))
46	43, 45	eqeq12d 2744	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) ↔ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))))
47	46	rspcv 3604	. . . 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 406	. 2 ⊢ ((𝜑 ∧ ∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))) → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)))
50	41, 49	impbida 800	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 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3057 ∃wrex 3066 ∖ cdif 3942 {csn 4624 ‘cfv 6542 (class class class)co 7414 Basecbs 17173 +_gcplusg 17226 Scalarcsca 17229 ·_𝑠 cvsca 17230 0_gc0g 17414 LSpanclspn 20848 HLchlt 38816 LHypclh 39451 DVecHcdvh 40545 LCDualclcd 41053 HDMapchdma 41259

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-riotaBAD 38419

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-tpos 8225 df-undef 8272 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-n0 12497 df-z 12583 df-uz 12847 df-fz 13511 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-0g 17416 df-mre 17559 df-mrc 17560 df-acs 17562 df-proset 18280 df-poset 18298 df-plt 18315 df-lub 18331 df-glb 18332 df-join 18333 df-meet 18334 df-p0 18410 df-p1 18411 df-lat 18417 df-clat 18484 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-submnd 18734 df-grp 18886 df-minusg 18887 df-sbg 18888 df-subg 19071 df-cntz 19261 df-oppg 19290 df-lsm 19584 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-oppr 20266 df-dvdsr 20289 df-unit 20290 df-invr 20320 df-dvr 20333 df-drng 20619 df-lmod 20738 df-lss 20809 df-lsp 20849 df-lvec 20981 df-lsatoms 38442 df-lshyp 38443 df-lcv 38485 df-lfl 38524 df-lkr 38552 df-ldual 38590 df-oposet 38642 df-ol 38644 df-oml 38645 df-covers 38732 df-ats 38733 df-atl 38764 df-cvlat 38788 df-hlat 38817 df-llines 38965 df-lplanes 38966 df-lvols 38967 df-lines 38968 df-psubsp 38970 df-pmap 38971 df-padd 39263 df-lhyp 39455 df-laut 39456 df-ldil 39571 df-ltrn 39572 df-trl 39626 df-tgrp 40210 df-tendo 40222 df-edring 40224 df-dveca 40470 df-disoa 40496 df-dvech 40546 df-dib 40606 df-dic 40640 df-dih 40696 df-doch 40815 df-djh 40862 df-lcdual 41054 df-mapd 41092 df-hvmap 41224 df-hdmap1 41260 df-hdmap 41261

This theorem is referenced by: hdmap14lem13 41347

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