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Theorem hlhilphllem 39099

Description: Lemma for hlhil 24049. (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 39076	. 2 ⊢ (𝜑 → 𝑉 = (Base‘𝑈))
7		hlhilphllem.a	. . 3 ⊢ + = (+_g‘𝐿)
8	1, 2, 3, 4, 7	hlhilplus 39077	. 2 ⊢ (𝜑 → + = (+_g‘𝑈))
9		hlhilphllem.s	. . 3 ⊢ · = ( ·_𝑠 ‘𝐿)
10	1, 4, 9, 2, 3	hlhilvsca 39087	. 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 39095	. 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 39082	. 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐹))
20		hlhilphllem.p	. . 3 ⊢ ⨣ = (+_g‘𝑅)
21	1, 4, 17, 2, 15, 3, 20	hlhilsplus2 39083	. 2 ⊢ (𝜑 → ⨣ = (+_g‘𝐹))
22		hlhilphllem.t	. . 3 ⊢ × = (._r‘𝑅)
23	1, 4, 17, 2, 15, 3, 22	hlhilsmul2 39084	. 2 ⊢ (𝜑 → × = (._r‘𝐹))
24		hlhilphllem.g	. . 3 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
25	1, 2, 15, 24, 3	hlhilnvl 39090	. 2 ⊢ (𝜑 → 𝐺 = (*_𝑟‘𝐹))
26		hlhilphllem.q	. . 3 ⊢ 𝑄 = (0_g‘𝑅)
27	1, 4, 17, 2, 15, 3, 26	hlhils0 39085	. 2 ⊢ (𝜑 → 𝑄 = (0_g‘𝐹))
28	1, 2, 3	hlhillvec 39091	. 2 ⊢ (𝜑 → 𝑈 ∈ LVec)
29	1, 2, 3, 15	hlhilsrng 39094	. 2 ⊢ (𝜑 → 𝐹 ∈ *-Ring)
30		hlhilphllem.j	. . . 4 ⊢ 𝐽 = ((HDMap‘𝐾)‘𝑊)
31	3	3ad2ant1 1129	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
32		simp2 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑎 ∈ 𝑉)
33		simp3 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ 𝑉)
34	1, 4, 5, 30, 2, 31, 11, 32, 33	hlhilipval 39089	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎 , 𝑏) = ((𝐽‘𝑏)‘𝑎))
35	1, 4, 5, 17, 18, 30, 31, 32, 33	hdmapipcl 39045	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝐽‘𝑏)‘𝑎) ∈ 𝐵)
36	34, 35	eqeltrd 2916	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎 , 𝑏) ∈ 𝐵)
37	3	3ad2ant1 1129	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
38		simp31 1205	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝑎 ∈ 𝑉)
39		simp32 1206	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝑏 ∈ 𝑉)
40		simp33 1207	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝑐 ∈ 𝑉)
41		simp2 1133	. . . 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 39046	. . 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 38250	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ LMod)
44	43	3ad2ant1 1129	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝐿 ∈ LMod)
45	5, 17, 9, 18	lmodvscl 19654	. . . . . 6 ⊢ ((𝐿 ∈ LMod ∧ 𝑑 ∈ 𝐵 ∧ 𝑎 ∈ 𝑉) → (𝑑 · 𝑎) ∈ 𝑉)
46	44, 41, 38, 45	syl3anc 1367	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑑 · 𝑎) ∈ 𝑉)
47	5, 7	lmodvacl 19651	. . . . 5 ⊢ ((𝐿 ∈ LMod ∧ (𝑑 · 𝑎) ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝑑 · 𝑎) + 𝑏) ∈ 𝑉)
48	44, 46, 39, 47	syl3anc 1367	. . . 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 39089	. . 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 39089	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑎 , 𝑐) = ((𝐽‘𝑐)‘𝑎))
51	50	oveq2d 7175	. . . 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 39089	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑏 , 𝑐) = ((𝐽‘𝑐)‘𝑏))
53	51, 52	oveq12d 7177	. . 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 2869	. 2 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (((𝑑 · 𝑎) + 𝑏) , 𝑐) = ((𝑑 × (𝑎 , 𝑐)) ⨣ (𝑏 , 𝑐)))
55	3	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
56		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → 𝑎 ∈ 𝑉)
57	1, 4, 5, 30, 2, 55, 11, 56, 56	hlhilipval 39089	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → (𝑎 , 𝑎) = ((𝐽‘𝑎)‘𝑎))
58	57	eqeq1d 2826	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → ((𝑎 , 𝑎) = 𝑄 ↔ ((𝐽‘𝑎)‘𝑎) = 𝑄))
59	1, 4, 5, 13, 17, 26, 30, 55, 56	hdmapip0 39055	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → (((𝐽‘𝑎)‘𝑎) = 𝑄 ↔ 𝑎 = 0 ))
60	58, 59	bitrd 281	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → ((𝑎 , 𝑎) = 𝑄 ↔ 𝑎 = 0 ))
61	60	biimp3a 1465	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ (𝑎 , 𝑎) = 𝑄) → 𝑎 = 0 )
62	1, 4, 5, 30, 24, 31, 32, 33	hdmapg 39070	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐺‘((𝐽‘𝑏)‘𝑎)) = ((𝐽‘𝑎)‘𝑏))
63	34	fveq2d 6677	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐺‘(𝑎 , 𝑏)) = (𝐺‘((𝐽‘𝑏)‘𝑎)))
64	1, 4, 5, 30, 2, 31, 11, 33, 32	hlhilipval 39089	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑏 , 𝑎) = ((𝐽‘𝑎)‘𝑏))
65	62, 63, 64	3eqtr4d 2869	. 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 20801	1 ⊢ (𝜑 → 𝑈 ∈ PreHil)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 ∈ cmpo 7161 Basecbs 16486 +_gcplusg 16568 ._rcmulr 16569 Scalarcsca 16571 ·_𝑠 cvsca 16572 ·_𝑖cip 16573 0_gc0g 16716 LModclmod 19637 PreHilcphl 20771 HLchlt 36490 LHypclh 37124 DVecHcdvh 38218 HDMapchdma 38932 HGMapchg 39023 HLHilchlh 39072

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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-riotaBAD 36093

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-ot 4579 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-tpos 7895 df-undef 7942 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-ip 16586 df-0g 16718 df-mre 16860 df-mrc 16861 df-acs 16863 df-proset 17541 df-poset 17559 df-plt 17571 df-lub 17587 df-glb 17588 df-join 17589 df-meet 17590 df-p0 17652 df-p1 17653 df-lat 17659 df-clat 17721 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-mhm 17959 df-submnd 17960 df-grp 18109 df-minusg 18110 df-sbg 18111 df-subg 18279 df-ghm 18359 df-cntz 18450 df-oppg 18477 df-lsm 18764 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-oppr 19376 df-dvdsr 19394 df-unit 19395 df-invr 19425 df-dvr 19436 df-rnghom 19470 df-drng 19507 df-subrg 19536 df-staf 19619 df-srng 19620 df-lmod 19639 df-lss 19707 df-lsp 19747 df-lmhm 19797 df-lvec 19878 df-sra 19947 df-rgmod 19948 df-phl 20773 df-lsatoms 36116 df-lshyp 36117 df-lcv 36159 df-lfl 36198 df-lkr 36226 df-ldual 36264 df-oposet 36316 df-ol 36318 df-oml 36319 df-covers 36406 df-ats 36407 df-atl 36438 df-cvlat 36462 df-hlat 36491 df-llines 36638 df-lplanes 36639 df-lvols 36640 df-lines 36641 df-psubsp 36643 df-pmap 36644 df-padd 36936 df-lhyp 37128 df-laut 37129 df-ldil 37244 df-ltrn 37245 df-trl 37299 df-tgrp 37883 df-tendo 37895 df-edring 37897 df-dveca 38143 df-disoa 38169 df-dvech 38219 df-dib 38279 df-dic 38313 df-dih 38369 df-doch 38488 df-djh 38535 df-lcdual 38727 df-mapd 38765 df-hvmap 38897 df-hdmap1 38933 df-hdmap 38934 df-hgmap 39024 df-hlhil 39073

This theorem is referenced by: hlhilhillem 39100

W3C validator