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Theorem hlhilphllem 42064

Description: Lemma for hlhil 25376. (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 42041	. 2 ⊢ (𝜑 → 𝑉 = (Base‘𝑈))
7		hlhilphllem.a	. . 3 ⊢ + = (+_g‘𝐿)
8	1, 2, 3, 4, 7	hlhilplus 42042	. 2 ⊢ (𝜑 → + = (+_g‘𝑈))
9		hlhilphllem.s	. . 3 ⊢ · = ( ·_𝑠 ‘𝐿)
10	1, 4, 9, 2, 3	hlhilvsca 42052	. 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 42060	. 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 42047	. 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐹))
20		hlhilphllem.p	. . 3 ⊢ ⨣ = (+_g‘𝑅)
21	1, 4, 17, 2, 15, 3, 20	hlhilsplus2 42048	. 2 ⊢ (𝜑 → ⨣ = (+_g‘𝐹))
22		hlhilphllem.t	. . 3 ⊢ × = (._r‘𝑅)
23	1, 4, 17, 2, 15, 3, 22	hlhilsmul2 42049	. 2 ⊢ (𝜑 → × = (._r‘𝐹))
24		hlhilphllem.g	. . 3 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
25	1, 2, 15, 24, 3	hlhilnvl 42055	. 2 ⊢ (𝜑 → 𝐺 = (*_𝑟‘𝐹))
26		hlhilphllem.q	. . 3 ⊢ 𝑄 = (0_g‘𝑅)
27	1, 4, 17, 2, 15, 3, 26	hlhils0 42050	. 2 ⊢ (𝜑 → 𝑄 = (0_g‘𝐹))
28	1, 2, 3	hlhillvec 42056	. 2 ⊢ (𝜑 → 𝑈 ∈ LVec)
29	1, 2, 3, 15	hlhilsrng 42059	. 2 ⊢ (𝜑 → 𝐹 ∈ *-Ring)
30		hlhilphllem.j	. . . 4 ⊢ 𝐽 = ((HDMap‘𝐾)‘𝑊)
31	3	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
32		simp2 1137	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑎 ∈ 𝑉)
33		simp3 1138	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ 𝑉)
34	1, 4, 5, 30, 2, 31, 11, 32, 33	hlhilipval 42054	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎 , 𝑏) = ((𝐽‘𝑏)‘𝑎))
35	1, 4, 5, 17, 18, 30, 31, 32, 33	hdmapipcl 42010	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝐽‘𝑏)‘𝑎) ∈ 𝐵)
36	34, 35	eqeltrd 2831	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎 , 𝑏) ∈ 𝐵)
37	3	3ad2ant1 1133	. . . 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 1137	. . . 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 42011	. . 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 41215	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ LMod)
44	43	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝐿 ∈ LMod)
45	5, 17, 9, 18	lmodvscl 20817	. . . . . 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 20814	. . . . 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 42054	. . 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 42054	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑎 , 𝑐) = ((𝐽‘𝑐)‘𝑎))
51	50	oveq2d 7368	. . . 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 42054	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑏 , 𝑐) = ((𝐽‘𝑐)‘𝑏))
53	51, 52	oveq12d 7370	. . 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 2776	. 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 42054	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → (𝑎 , 𝑎) = ((𝐽‘𝑎)‘𝑎))
58	57	eqeq1d 2733	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → ((𝑎 , 𝑎) = 𝑄 ↔ ((𝐽‘𝑎)‘𝑎) = 𝑄))
59	1, 4, 5, 13, 17, 26, 30, 55, 56	hdmapip0 42020	. . . 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 42035	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐺‘((𝐽‘𝑏)‘𝑎)) = ((𝐽‘𝑎)‘𝑏))
63	34	fveq2d 6832	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐺‘(𝑎 , 𝑏)) = (𝐺‘((𝐽‘𝑏)‘𝑎)))
64	1, 4, 5, 30, 2, 31, 11, 33, 32	hlhilipval 42054	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑏 , 𝑎) = ((𝐽‘𝑎)‘𝑏))
65	62, 63, 64	3eqtr4d 2776	. 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 21597	1 ⊢ (𝜑 → 𝑈 ∈ PreHil)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ‘cfv 6487 (class class class)co 7352 ∈ cmpo 7354 Basecbs 17126 +_gcplusg 17167 ._rcmulr 17168 Scalarcsca 17170 ·_𝑠 cvsca 17171 ·_𝑖cip 17172 0_gc0g 17349 LModclmod 20799 PreHilcphl 21567 HLchlt 39455 LHypclh 40089 DVecHcdvh 41183 HDMapchdma 41897 HGMapchg 41988 HLHilchlh 42037

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11068 ax-resscn 11069 ax-1cn 11070 ax-icn 11071 ax-addcl 11072 ax-addrcl 11073 ax-mulcl 11074 ax-mulrcl 11075 ax-mulcom 11076 ax-addass 11077 ax-mulass 11078 ax-distr 11079 ax-i2m1 11080 ax-1ne0 11081 ax-1rid 11082 ax-rnegex 11083 ax-rrecex 11084 ax-cnre 11085 ax-pre-lttri 11086 ax-pre-lttrn 11087 ax-pre-ltadd 11088 ax-pre-mulgt0 11089 ax-riotaBAD 39058

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-ot 4584 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-om 7803 df-1st 7927 df-2nd 7928 df-tpos 8162 df-undef 8209 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-er 8628 df-map 8758 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-pnf 11154 df-mnf 11155 df-xr 11156 df-ltxr 11157 df-le 11158 df-sub 11352 df-neg 11353 df-nn 12132 df-2 12194 df-3 12195 df-4 12196 df-5 12197 df-6 12198 df-7 12199 df-8 12200 df-n0 12388 df-z 12475 df-uz 12739 df-fz 13414 df-struct 17064 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17127 df-ress 17148 df-plusg 17180 df-mulr 17181 df-starv 17182 df-sca 17183 df-vsca 17184 df-ip 17185 df-0g 17351 df-mre 17494 df-mrc 17495 df-acs 17497 df-proset 18206 df-poset 18225 df-plt 18240 df-lub 18256 df-glb 18257 df-join 18258 df-meet 18259 df-p0 18335 df-p1 18336 df-lat 18344 df-clat 18411 df-mgm 18554 df-sgrp 18633 df-mnd 18649 df-mhm 18697 df-submnd 18698 df-grp 18855 df-minusg 18856 df-sbg 18857 df-subg 19042 df-ghm 19131 df-cntz 19235 df-oppg 19264 df-lsm 19554 df-cmn 19700 df-abl 19701 df-mgp 20065 df-rng 20077 df-ur 20106 df-ring 20159 df-oppr 20261 df-dvdsr 20281 df-unit 20282 df-invr 20312 df-dvr 20325 df-rhm 20396 df-nzr 20434 df-subrg 20491 df-rlreg 20615 df-domn 20616 df-drng 20652 df-staf 20760 df-srng 20761 df-lmod 20801 df-lss 20871 df-lsp 20911 df-lmhm 20962 df-lvec 21043 df-sra 21113 df-rgmod 21114 df-phl 21569 df-lsatoms 39081 df-lshyp 39082 df-lcv 39124 df-lfl 39163 df-lkr 39191 df-ldual 39229 df-oposet 39281 df-ol 39283 df-oml 39284 df-covers 39371 df-ats 39372 df-atl 39403 df-cvlat 39427 df-hlat 39456 df-llines 39603 df-lplanes 39604 df-lvols 39605 df-lines 39606 df-psubsp 39608 df-pmap 39609 df-padd 39901 df-lhyp 40093 df-laut 40094 df-ldil 40209 df-ltrn 40210 df-trl 40264 df-tgrp 40848 df-tendo 40860 df-edring 40862 df-dveca 41108 df-disoa 41134 df-dvech 41184 df-dib 41244 df-dic 41278 df-dih 41334 df-doch 41453 df-djh 41500 df-lcdual 41692 df-mapd 41730 df-hvmap 41862 df-hdmap1 41898 df-hdmap 41899 df-hgmap 41989 df-hlhil 42038

This theorem is referenced by: hlhilhillem 42065

W3C validator