hdmap14lem12 - Mathbox for Norm Megill

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Theorem hdmap14lem12 43030

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 2760	. . . . . 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 1151	. . . . . 6 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15		simp3 1156	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → 𝑦 ∈ (𝑉 ∖ { 0 }))
16	15	eldifad 4051	. . . . . 6 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → 𝑦 ∈ 𝑉)
17		hdmap14lem12.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
18	17	3ad2ant1 1151	. . . . . 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 43018	. . . . 5 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → ∃𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦)))
20		simp3 1156	. . . . . . 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 2760	. . . . . . . . 9 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
22		hdmap14lem12.o	. . . . . . . . 9 ⊢ 0 = (0_g‘𝑈)
23		eqid 2760	. . . . . . . . 9 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
24		eqid 2760	. . . . . . . . 9 ⊢ (+_g‘𝐶) = (+_g‘𝐶)
25		simp11 1222	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → 𝜑)
26	25, 13	syl 18	. . . . . . . . 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 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → 𝑋 ∈ (𝑉 ∖ { 0 }))
29		simp13 1224	. . . . . . . . 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 18	. . . . . . . . 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 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → 𝐺 ∈ 𝐴)
33		simp2 1155	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → 𝑔 ∈ 𝐴)
34		simp12 1223	. . . . . . . . 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 43029	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · 𝑦)) = (𝑔 ∙ (𝑆‘𝑦))) → 𝐺 = 𝑔)
36	35	oveq1d 7436	. . . . . . 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 2798	. . . . . 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 3167	. . . . 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 16	. . . 4 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ∧ 𝑦 ∈ (𝑉 ∖ { 0 })) → (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))
40	39	3expia 1139	. . 3 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) → (𝑦 ∈ (𝑉 ∖ { 0 }) → (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))))
41	40	ralrimiv 3153	. 2 ⊢ ((𝜑 ∧ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) → ∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))
42		oveq2 7429	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝐹 · 𝑦) = (𝐹 · 𝑋))
43	42	fveq2d 6890	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑆‘(𝐹 · 𝑦)) = (𝑆‘(𝐹 · 𝑋)))
44		fveq2 6886	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝑆‘𝑦) = (𝑆‘𝑋))
45	44	oveq2d 7437	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝐺 ∙ (𝑆‘𝑦)) = (𝐺 ∙ (𝑆‘𝑋)))
46	43, 45	eqeq12d 2776	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) ↔ (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))))
47	46	rspcv 3572	. . . 4 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → (∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))))
48	27, 47	syl 18	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)) → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))))
49	48	imp 412	. 2 ⊢ ((𝜑 ∧ ∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))) → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)))
50	41, 49	impbida 813	1 ⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ↔ ∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ∀wral 3076 ∃wrex 3086 ∖ cdif 4045 {csn 4584 ‘cfv 6539 (class class class)co 7421 Basecbs 17393 +_gcplusg 17434 Scalarcsca 17437 ·_𝑠 cvsca 17438 0_gc0g 17616 LSpanclspn 21252 HLchlt 40501 LHypclh 41135 DVecHcdvh 42229 LCDualclcd 42737 HDMapchdma 42943

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281 ax-riotaBAD 40104

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-ot 4858 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-of 7694 df-om 7879 df-1st 8002 df-2nd 8003 df-tpos 8247 df-undef 8294 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-2o 8481 df-er 8721 df-map 8853 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-nn 12340 df-2 12409 df-3 12410 df-4 12411 df-5 12412 df-6 12413 df-n0 12611 df-z 12698 df-uz 12970 df-fz 13644 df-struct 17331 df-sets 17348 df-slot 17366 df-ndx 17378 df-base 17394 df-ress 17415 df-plusg 17447 df-mulr 17448 df-sca 17450 df-vsca 17451 df-0g 17618 df-mre 17762 df-mrc 17763 df-acs 17765 df-proset 18474 df-poset 18493 df-plt 18508 df-lub 18524 df-glb 18525 df-join 18526 df-meet 18527 df-p0 18603 df-p1 18604 df-lat 18612 df-clat 18679 df-mgm 18822 df-sgrp 18914 df-mnd 18930 df-submnd 18985 df-grp 19153 df-minusg 19154 df-sbg 19155 df-subg 19339 df-cntz 19537 df-oppg 19566 df-lsm 19856 df-cmn 20002 df-abl 20003 df-mgp 20367 df-rng 20381 df-ur 20414 df-ring 20467 df-oppr 20573 df-dvdsr 20593 df-unit 20594 df-invr 20624 df-dvr 20637 df-nzr 20769 df-rlreg 20952 df-domn 20953 df-drng 20988 df-lmod 21143 df-lss 21213 df-lsp 21253 df-lvec 21384 df-lsatoms 40127 df-lshyp 40128 df-lcv 40170 df-lfl 40209 df-lkr 40237 df-ldual 40275 df-oposet 40327 df-ol 40329 df-oml 40330 df-covers 40417 df-ats 40418 df-atl 40449 df-cvlat 40473 df-hlat 40502 df-llines 40649 df-lplanes 40650 df-lvols 40651 df-lines 40652 df-psubsp 40654 df-pmap 40655 df-padd 40947 df-lhyp 41139 df-laut 41140 df-ldil 41255 df-ltrn 41256 df-trl 41310 df-tgrp 41894 df-tendo 41906 df-edring 41908 df-dveca 42154 df-disoa 42180 df-dvech 42230 df-dib 42290 df-dic 42324 df-dih 42380 df-doch 42499 df-djh 42546 df-lcdual 42738 df-mapd 42776 df-hvmap 42908 df-hdmap1 42944 df-hdmap 42945

This theorem is used by: hdmap14lem13 43031

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