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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlhilphllem | Structured version Visualization version GIF version |
Description: Lemma for hlhil 25415. (Contributed by NM, 23-Jun-2015.) |
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 = (0g‘𝑅) |
hlhilphllem.z | ⊢ 0 = (0g‘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)) |
Ref | Expression |
---|---|
hlhilphllem | ⊢ (𝜑 → 𝑈 ∈ PreHil) |
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 41539 | . 2 ⊢ (𝜑 → 𝑉 = (Base‘𝑈)) |
7 | hlhilphllem.a | . . 3 ⊢ + = (+g‘𝐿) | |
8 | 1, 2, 3, 4, 7 | hlhilplus 41540 | . 2 ⊢ (𝜑 → + = (+g‘𝑈)) |
9 | hlhilphllem.s | . . 3 ⊢ · = ( ·𝑠 ‘𝐿) | |
10 | 1, 4, 9, 2, 3 | hlhilvsca 41554 | . 2 ⊢ (𝜑 → · = ( ·𝑠 ‘𝑈)) |
11 | hlhilphllem.i | . . 3 ⊢ , = (·𝑖‘𝑈) | |
12 | 11 | a1i 11 | . 2 ⊢ (𝜑 → , = (·𝑖‘𝑈)) |
13 | hlhilphllem.z | . . 3 ⊢ 0 = (0g‘𝐿) | |
14 | 1, 4, 2, 3, 13 | hlhil0 41562 | . 2 ⊢ (𝜑 → 0 = (0g‘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 41549 | . 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐹)) |
20 | hlhilphllem.p | . . 3 ⊢ ⨣ = (+g‘𝑅) | |
21 | 1, 4, 17, 2, 15, 3, 20 | hlhilsplus2 41550 | . 2 ⊢ (𝜑 → ⨣ = (+g‘𝐹)) |
22 | hlhilphllem.t | . . 3 ⊢ × = (.r‘𝑅) | |
23 | 1, 4, 17, 2, 15, 3, 22 | hlhilsmul2 41551 | . 2 ⊢ (𝜑 → × = (.r‘𝐹)) |
24 | hlhilphllem.g | . . 3 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) | |
25 | 1, 2, 15, 24, 3 | hlhilnvl 41557 | . 2 ⊢ (𝜑 → 𝐺 = (*𝑟‘𝐹)) |
26 | hlhilphllem.q | . . 3 ⊢ 𝑄 = (0g‘𝑅) | |
27 | 1, 4, 17, 2, 15, 3, 26 | hlhils0 41552 | . 2 ⊢ (𝜑 → 𝑄 = (0g‘𝐹)) |
28 | 1, 2, 3 | hlhillvec 41558 | . 2 ⊢ (𝜑 → 𝑈 ∈ LVec) |
29 | 1, 2, 3, 15 | hlhilsrng 41561 | . 2 ⊢ (𝜑 → 𝐹 ∈ *-Ring) |
30 | hlhilphllem.j | . . . 4 ⊢ 𝐽 = ((HDMap‘𝐾)‘𝑊) | |
31 | 3 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
32 | simp2 1134 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑎 ∈ 𝑉) | |
33 | simp3 1135 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ 𝑉) | |
34 | 1, 4, 5, 30, 2, 31, 11, 32, 33 | hlhilipval 41556 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎 , 𝑏) = ((𝐽‘𝑏)‘𝑎)) |
35 | 1, 4, 5, 17, 18, 30, 31, 32, 33 | hdmapipcl 41508 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝐽‘𝑏)‘𝑎) ∈ 𝐵) |
36 | 34, 35 | eqeltrd 2825 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎 , 𝑏) ∈ 𝐵) |
37 | 3 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
38 | simp31 1206 | . . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝑎 ∈ 𝑉) | |
39 | simp32 1207 | . . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝑏 ∈ 𝑉) | |
40 | simp33 1208 | . . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝑐 ∈ 𝑉) | |
41 | simp2 1134 | . . . 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 41509 | . . 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 40713 | . . . . . 6 ⊢ (𝜑 → 𝐿 ∈ LMod) |
44 | 43 | 3ad2ant1 1130 | . . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝐿 ∈ LMod) |
45 | 5, 17, 9, 18 | lmodvscl 20773 | . . . . . 6 ⊢ ((𝐿 ∈ LMod ∧ 𝑑 ∈ 𝐵 ∧ 𝑎 ∈ 𝑉) → (𝑑 · 𝑎) ∈ 𝑉) |
46 | 44, 41, 38, 45 | syl3anc 1368 | . . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑑 · 𝑎) ∈ 𝑉) |
47 | 5, 7 | lmodvacl 20770 | . . . . 5 ⊢ ((𝐿 ∈ LMod ∧ (𝑑 · 𝑎) ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝑑 · 𝑎) + 𝑏) ∈ 𝑉) |
48 | 44, 46, 39, 47 | syl3anc 1368 | . . . 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 41556 | . . 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 41556 | . . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑎 , 𝑐) = ((𝐽‘𝑐)‘𝑎)) |
51 | 50 | oveq2d 7435 | . . . 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 41556 | . . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑏 , 𝑐) = ((𝐽‘𝑐)‘𝑏)) |
53 | 51, 52 | oveq12d 7437 | . . 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 2775 | . 2 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (((𝑑 · 𝑎) + 𝑏) , 𝑐) = ((𝑑 × (𝑎 , 𝑐)) ⨣ (𝑏 , 𝑐))) |
55 | 3 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
56 | simpr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → 𝑎 ∈ 𝑉) | |
57 | 1, 4, 5, 30, 2, 55, 11, 56, 56 | hlhilipval 41556 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → (𝑎 , 𝑎) = ((𝐽‘𝑎)‘𝑎)) |
58 | 57 | eqeq1d 2727 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → ((𝑎 , 𝑎) = 𝑄 ↔ ((𝐽‘𝑎)‘𝑎) = 𝑄)) |
59 | 1, 4, 5, 13, 17, 26, 30, 55, 56 | hdmapip0 41518 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → (((𝐽‘𝑎)‘𝑎) = 𝑄 ↔ 𝑎 = 0 )) |
60 | 58, 59 | bitrd 278 | . . 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 41533 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐺‘((𝐽‘𝑏)‘𝑎)) = ((𝐽‘𝑎)‘𝑏)) |
63 | 34 | fveq2d 6900 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐺‘(𝑎 , 𝑏)) = (𝐺‘((𝐽‘𝑏)‘𝑎))) |
64 | 1, 4, 5, 30, 2, 31, 11, 33, 32 | hlhilipval 41556 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑏 , 𝑎) = ((𝐽‘𝑎)‘𝑏)) |
65 | 62, 63, 64 | 3eqtr4d 2775 | . 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 21603 | 1 ⊢ (𝜑 → 𝑈 ∈ PreHil) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ‘cfv 6549 (class class class)co 7419 ∈ cmpo 7421 Basecbs 17183 +gcplusg 17236 .rcmulr 17237 Scalarcsca 17239 ·𝑠 cvsca 17240 ·𝑖cip 17241 0gc0g 17424 LModclmod 20755 PreHilcphl 21573 HLchlt 38952 LHypclh 39587 DVecHcdvh 40681 HDMapchdma 41395 HGMapchg 41486 HLHilchlh 41535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-riotaBAD 38555 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-ot 4639 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-undef 8279 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-starv 17251 df-sca 17252 df-vsca 17253 df-ip 17254 df-0g 17426 df-mre 17569 df-mrc 17570 df-acs 17572 df-proset 18290 df-poset 18308 df-plt 18325 df-lub 18341 df-glb 18342 df-join 18343 df-meet 18344 df-p0 18420 df-p1 18421 df-lat 18427 df-clat 18494 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18743 df-submnd 18744 df-grp 18901 df-minusg 18902 df-sbg 18903 df-subg 19086 df-ghm 19176 df-cntz 19280 df-oppg 19309 df-lsm 19603 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-dvr 20352 df-rhm 20423 df-subrg 20520 df-drng 20638 df-staf 20737 df-srng 20738 df-lmod 20757 df-lss 20828 df-lsp 20868 df-lmhm 20919 df-lvec 21000 df-sra 21070 df-rgmod 21071 df-phl 21575 df-lsatoms 38578 df-lshyp 38579 df-lcv 38621 df-lfl 38660 df-lkr 38688 df-ldual 38726 df-oposet 38778 df-ol 38780 df-oml 38781 df-covers 38868 df-ats 38869 df-atl 38900 df-cvlat 38924 df-hlat 38953 df-llines 39101 df-lplanes 39102 df-lvols 39103 df-lines 39104 df-psubsp 39106 df-pmap 39107 df-padd 39399 df-lhyp 39591 df-laut 39592 df-ldil 39707 df-ltrn 39708 df-trl 39762 df-tgrp 40346 df-tendo 40358 df-edring 40360 df-dveca 40606 df-disoa 40632 df-dvech 40682 df-dib 40742 df-dic 40776 df-dih 40832 df-doch 40951 df-djh 40998 df-lcdual 41190 df-mapd 41228 df-hvmap 41360 df-hdmap1 41396 df-hdmap 41397 df-hgmap 41487 df-hlhil 41536 |
This theorem is referenced by: hlhilhillem 41567 |
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