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Theorem hlhilphllem 41965

Description: Lemma for hlhil 25477. (Contributed by NM, 23-Jun-2015.)

**Hypotheses**
Ref	Expression
hlhilphl.h	⊢ 𝐻 = (LHyp‘𝐾)
hlhilphllem.u	⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊)
hlhilphl.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hlhilphllem.f	⊢ 𝐹 = (Scalar‘𝑈)
hlhilphllem.l	⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊)
hlhilphllem.v	⊢ 𝑉 = (Base‘𝐿)
hlhilphllem.a	⊢ + = (+_g‘𝐿)
hlhilphllem.s	⊢ · = ( ·_𝑠 ‘𝐿)
hlhilphllem.r	⊢ 𝑅 = (Scalar‘𝐿)
hlhilphllem.b	⊢ 𝐵 = (Base‘𝑅)
hlhilphllem.p	⊢ ⨣ = (+_g‘𝑅)
hlhilphllem.t	⊢ × = (._r‘𝑅)
hlhilphllem.q	⊢ 𝑄 = (0_g‘𝑅)
hlhilphllem.z	⊢ 0 = (0_g‘𝐿)
hlhilphllem.i	⊢ , = (·_𝑖‘𝑈)
hlhilphllem.j	⊢ 𝐽 = ((HDMap‘𝐾)‘𝑊)
hlhilphllem.g	⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
hlhilphllem.e	⊢ 𝐸 = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝐽‘𝑦)‘𝑥))

**Assertion**
Ref	Expression
hlhilphllem	⊢ (𝜑 → 𝑈 ∈ PreHil)

Distinct variable groups: 𝑥,𝑦,𝐾 𝑥,𝑈 𝑥,𝑊,𝑦 𝜑,𝑥 𝑥,𝐽,𝑦 𝑥,𝑉,𝑦 Allowed substitution hints: 𝜑(𝑦) 𝐵(𝑥,𝑦) + (𝑥,𝑦) ⨣ (𝑥,𝑦) 𝑄(𝑥,𝑦) 𝑅(𝑥,𝑦) · (𝑥,𝑦) × (𝑥,𝑦) 𝑈(𝑦) 𝐸(𝑥,𝑦) 𝐹(𝑥,𝑦) 𝐺(𝑥,𝑦) 𝐻(𝑥,𝑦) , (𝑥,𝑦) 𝐿(𝑥,𝑦) 0 (𝑥,𝑦)

Proof of Theorem hlhilphllem Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hlhilphl.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		hlhilphllem.u	. . 3 ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊)
3		hlhilphl.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4		hlhilphllem.l	. . 3 ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊)
5		hlhilphllem.v	. . 3 ⊢ 𝑉 = (Base‘𝐿)
6	1, 2, 3, 4, 5	hlhilbase 41938	. 2 ⊢ (𝜑 → 𝑉 = (Base‘𝑈))
7		hlhilphllem.a	. . 3 ⊢ + = (+_g‘𝐿)
8	1, 2, 3, 4, 7	hlhilplus 41939	. 2 ⊢ (𝜑 → + = (+_g‘𝑈))
9		hlhilphllem.s	. . 3 ⊢ · = ( ·_𝑠 ‘𝐿)
10	1, 4, 9, 2, 3	hlhilvsca 41953	. 2 ⊢ (𝜑 → · = ( ·_𝑠 ‘𝑈))
11		hlhilphllem.i	. . 3 ⊢ , = (·_𝑖‘𝑈)
12	11	a1i 11	. 2 ⊢ (𝜑 → , = (·_𝑖‘𝑈))
13		hlhilphllem.z	. . 3 ⊢ 0 = (0_g‘𝐿)
14	1, 4, 2, 3, 13	hlhil0 41961	. 2 ⊢ (𝜑 → 0 = (0_g‘𝑈))
15		hlhilphllem.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑈)
16	15	a1i 11	. 2 ⊢ (𝜑 → 𝐹 = (Scalar‘𝑈))
17		hlhilphllem.r	. . 3 ⊢ 𝑅 = (Scalar‘𝐿)
18		hlhilphllem.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
19	1, 4, 17, 2, 15, 3, 18	hlhilsbase2 41948	. 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐹))
20		hlhilphllem.p	. . 3 ⊢ ⨣ = (+_g‘𝑅)
21	1, 4, 17, 2, 15, 3, 20	hlhilsplus2 41949	. 2 ⊢ (𝜑 → ⨣ = (+_g‘𝐹))
22		hlhilphllem.t	. . 3 ⊢ × = (._r‘𝑅)
23	1, 4, 17, 2, 15, 3, 22	hlhilsmul2 41950	. 2 ⊢ (𝜑 → × = (._r‘𝐹))
24		hlhilphllem.g	. . 3 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
25	1, 2, 15, 24, 3	hlhilnvl 41956	. 2 ⊢ (𝜑 → 𝐺 = (*_𝑟‘𝐹))
26		hlhilphllem.q	. . 3 ⊢ 𝑄 = (0_g‘𝑅)
27	1, 4, 17, 2, 15, 3, 26	hlhils0 41951	. 2 ⊢ (𝜑 → 𝑄 = (0_g‘𝐹))
28	1, 2, 3	hlhillvec 41957	. 2 ⊢ (𝜑 → 𝑈 ∈ LVec)
29	1, 2, 3, 15	hlhilsrng 41960	. 2 ⊢ (𝜑 → 𝐹 ∈ *-Ring)
30		hlhilphllem.j	. . . 4 ⊢ 𝐽 = ((HDMap‘𝐾)‘𝑊)
31	3	3ad2ant1 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
32		simp2 1138	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑎 ∈ 𝑉)
33		simp3 1139	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ 𝑉)
34	1, 4, 5, 30, 2, 31, 11, 32, 33	hlhilipval 41955	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎 , 𝑏) = ((𝐽‘𝑏)‘𝑎))
35	1, 4, 5, 17, 18, 30, 31, 32, 33	hdmapipcl 41907	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝐽‘𝑏)‘𝑎) ∈ 𝐵)
36	34, 35	eqeltrd 2841	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎 , 𝑏) ∈ 𝐵)
37	3	3ad2ant1 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
38		simp31 1210	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝑎 ∈ 𝑉)
39		simp32 1211	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝑏 ∈ 𝑉)
40		simp33 1212	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝑐 ∈ 𝑉)
41		simp2 1138	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝑑 ∈ 𝐵)
42	1, 4, 5, 7, 9, 17, 18, 20, 22, 30, 37, 38, 39, 40, 41	hdmapln1 41908	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → ((𝐽‘𝑐)‘((𝑑 · 𝑎) + 𝑏)) = ((𝑑 × ((𝐽‘𝑐)‘𝑎)) ⨣ ((𝐽‘𝑐)‘𝑏)))
43	1, 4, 3	dvhlmod 41112	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ LMod)
44	43	3ad2ant1 1134	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝐿 ∈ LMod)
45	5, 17, 9, 18	lmodvscl 20876	. . . . . 6 ⊢ ((𝐿 ∈ LMod ∧ 𝑑 ∈ 𝐵 ∧ 𝑎 ∈ 𝑉) → (𝑑 · 𝑎) ∈ 𝑉)
46	44, 41, 38, 45	syl3anc 1373	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑑 · 𝑎) ∈ 𝑉)
47	5, 7	lmodvacl 20873	. . . . 5 ⊢ ((𝐿 ∈ LMod ∧ (𝑑 · 𝑎) ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝑑 · 𝑎) + 𝑏) ∈ 𝑉)
48	44, 46, 39, 47	syl3anc 1373	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → ((𝑑 · 𝑎) + 𝑏) ∈ 𝑉)
49	1, 4, 5, 30, 2, 37, 11, 48, 40	hlhilipval 41955	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (((𝑑 · 𝑎) + 𝑏) , 𝑐) = ((𝐽‘𝑐)‘((𝑑 · 𝑎) + 𝑏)))
50	1, 4, 5, 30, 2, 37, 11, 38, 40	hlhilipval 41955	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑎 , 𝑐) = ((𝐽‘𝑐)‘𝑎))
51	50	oveq2d 7447	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑑 × (𝑎 , 𝑐)) = (𝑑 × ((𝐽‘𝑐)‘𝑎)))
52	1, 4, 5, 30, 2, 37, 11, 39, 40	hlhilipval 41955	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑏 , 𝑐) = ((𝐽‘𝑐)‘𝑏))
53	51, 52	oveq12d 7449	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → ((𝑑 × (𝑎 , 𝑐)) ⨣ (𝑏 , 𝑐)) = ((𝑑 × ((𝐽‘𝑐)‘𝑎)) ⨣ ((𝐽‘𝑐)‘𝑏)))
54	42, 49, 53	3eqtr4d 2787	. 2 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (((𝑑 · 𝑎) + 𝑏) , 𝑐) = ((𝑑 × (𝑎 , 𝑐)) ⨣ (𝑏 , 𝑐)))
55	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
56		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → 𝑎 ∈ 𝑉)
57	1, 4, 5, 30, 2, 55, 11, 56, 56	hlhilipval 41955	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → (𝑎 , 𝑎) = ((𝐽‘𝑎)‘𝑎))
58	57	eqeq1d 2739	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → ((𝑎 , 𝑎) = 𝑄 ↔ ((𝐽‘𝑎)‘𝑎) = 𝑄))
59	1, 4, 5, 13, 17, 26, 30, 55, 56	hdmapip0 41917	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → (((𝐽‘𝑎)‘𝑎) = 𝑄 ↔ 𝑎 = 0 ))
60	58, 59	bitrd 279	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → ((𝑎 , 𝑎) = 𝑄 ↔ 𝑎 = 0 ))
61	60	biimp3a 1471	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ (𝑎 , 𝑎) = 𝑄) → 𝑎 = 0 )
62	1, 4, 5, 30, 24, 31, 32, 33	hdmapg 41932	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐺‘((𝐽‘𝑏)‘𝑎)) = ((𝐽‘𝑎)‘𝑏))
63	34	fveq2d 6910	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐺‘(𝑎 , 𝑏)) = (𝐺‘((𝐽‘𝑏)‘𝑎)))
64	1, 4, 5, 30, 2, 31, 11, 33, 32	hlhilipval 41955	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑏 , 𝑎) = ((𝐽‘𝑎)‘𝑏))
65	62, 63, 64	3eqtr4d 2787	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐺‘(𝑎 , 𝑏)) = (𝑏 , 𝑎))
66	6, 8, 10, 12, 14, 16, 19, 21, 23, 25, 27, 28, 29, 36, 54, 61, 65	isphld 21672	1 ⊢ (𝜑 → 𝑈 ∈ PreHil)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 Basecbs 17247 +_gcplusg 17297 ._rcmulr 17298 Scalarcsca 17300 ·_𝑠 cvsca 17301 ·_𝑖cip 17302 0_gc0g 17484 LModclmod 20858 PreHilcphl 21642 HLchlt 39351 LHypclh 39986 DVecHcdvh 41080 HDMapchdma 41794 HGMapchg 41885 HLHilchlh 41934

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-riotaBAD 38954

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-undef 8298 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-0g 17486 df-mre 17629 df-mrc 17630 df-acs 17632 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-clat 18544 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-ghm 19231 df-cntz 19335 df-oppg 19364 df-lsm 19654 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-rhm 20472 df-nzr 20513 df-subrg 20570 df-rlreg 20694 df-domn 20695 df-drng 20731 df-staf 20840 df-srng 20841 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lmhm 21021 df-lvec 21102 df-sra 21172 df-rgmod 21173 df-phl 21644 df-lsatoms 38977 df-lshyp 38978 df-lcv 39020 df-lfl 39059 df-lkr 39087 df-ldual 39125 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-llines 39500 df-lplanes 39501 df-lvols 39502 df-lines 39503 df-psubsp 39505 df-pmap 39506 df-padd 39798 df-lhyp 39990 df-laut 39991 df-ldil 40106 df-ltrn 40107 df-trl 40161 df-tgrp 40745 df-tendo 40757 df-edring 40759 df-dveca 41005 df-disoa 41031 df-dvech 41081 df-dib 41141 df-dic 41175 df-dih 41231 df-doch 41350 df-djh 41397 df-lcdual 41589 df-mapd 41627 df-hvmap 41759 df-hdmap1 41795 df-hdmap 41796 df-hgmap 41886 df-hlhil 41935

This theorem is referenced by: hlhilhillem 41966

W3C validator