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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlhilphllem | Structured version Visualization version GIF version |
Description: Lemma for hlhil 25491. (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 41919 | . 2 ⊢ (𝜑 → 𝑉 = (Base‘𝑈)) |
7 | hlhilphllem.a | . . 3 ⊢ + = (+g‘𝐿) | |
8 | 1, 2, 3, 4, 7 | hlhilplus 41920 | . 2 ⊢ (𝜑 → + = (+g‘𝑈)) |
9 | hlhilphllem.s | . . 3 ⊢ · = ( ·𝑠 ‘𝐿) | |
10 | 1, 4, 9, 2, 3 | hlhilvsca 41934 | . 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 41942 | . 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 41929 | . 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐹)) |
20 | hlhilphllem.p | . . 3 ⊢ ⨣ = (+g‘𝑅) | |
21 | 1, 4, 17, 2, 15, 3, 20 | hlhilsplus2 41930 | . 2 ⊢ (𝜑 → ⨣ = (+g‘𝐹)) |
22 | hlhilphllem.t | . . 3 ⊢ × = (.r‘𝑅) | |
23 | 1, 4, 17, 2, 15, 3, 22 | hlhilsmul2 41931 | . 2 ⊢ (𝜑 → × = (.r‘𝐹)) |
24 | hlhilphllem.g | . . 3 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) | |
25 | 1, 2, 15, 24, 3 | hlhilnvl 41937 | . 2 ⊢ (𝜑 → 𝐺 = (*𝑟‘𝐹)) |
26 | hlhilphllem.q | . . 3 ⊢ 𝑄 = (0g‘𝑅) | |
27 | 1, 4, 17, 2, 15, 3, 26 | hlhils0 41932 | . 2 ⊢ (𝜑 → 𝑄 = (0g‘𝐹)) |
28 | 1, 2, 3 | hlhillvec 41938 | . 2 ⊢ (𝜑 → 𝑈 ∈ LVec) |
29 | 1, 2, 3, 15 | hlhilsrng 41941 | . 2 ⊢ (𝜑 → 𝐹 ∈ *-Ring) |
30 | hlhilphllem.j | . . . 4 ⊢ 𝐽 = ((HDMap‘𝐾)‘𝑊) | |
31 | 3 | 3ad2ant1 1132 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
32 | simp2 1136 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑎 ∈ 𝑉) | |
33 | simp3 1137 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ 𝑉) | |
34 | 1, 4, 5, 30, 2, 31, 11, 32, 33 | hlhilipval 41936 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎 , 𝑏) = ((𝐽‘𝑏)‘𝑎)) |
35 | 1, 4, 5, 17, 18, 30, 31, 32, 33 | hdmapipcl 41888 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝐽‘𝑏)‘𝑎) ∈ 𝐵) |
36 | 34, 35 | eqeltrd 2839 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎 , 𝑏) ∈ 𝐵) |
37 | 3 | 3ad2ant1 1132 | . . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
38 | simp31 1208 | . . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝑎 ∈ 𝑉) | |
39 | simp32 1209 | . . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝑏 ∈ 𝑉) | |
40 | simp33 1210 | . . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝑐 ∈ 𝑉) | |
41 | simp2 1136 | . . . 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 41889 | . . 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 41093 | . . . . . 6 ⊢ (𝜑 → 𝐿 ∈ LMod) |
44 | 43 | 3ad2ant1 1132 | . . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝐿 ∈ LMod) |
45 | 5, 17, 9, 18 | lmodvscl 20893 | . . . . . 6 ⊢ ((𝐿 ∈ LMod ∧ 𝑑 ∈ 𝐵 ∧ 𝑎 ∈ 𝑉) → (𝑑 · 𝑎) ∈ 𝑉) |
46 | 44, 41, 38, 45 | syl3anc 1370 | . . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑑 · 𝑎) ∈ 𝑉) |
47 | 5, 7 | lmodvacl 20890 | . . . . 5 ⊢ ((𝐿 ∈ LMod ∧ (𝑑 · 𝑎) ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝑑 · 𝑎) + 𝑏) ∈ 𝑉) |
48 | 44, 46, 39, 47 | syl3anc 1370 | . . . 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 41936 | . . 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 41936 | . . . . 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 41936 | . . . 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 2785 | . 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 41936 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → (𝑎 , 𝑎) = ((𝐽‘𝑎)‘𝑎)) |
58 | 57 | eqeq1d 2737 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → ((𝑎 , 𝑎) = 𝑄 ↔ ((𝐽‘𝑎)‘𝑎) = 𝑄)) |
59 | 1, 4, 5, 13, 17, 26, 30, 55, 56 | hdmapip0 41898 | . . . 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 1468 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ (𝑎 , 𝑎) = 𝑄) → 𝑎 = 0 ) |
62 | 1, 4, 5, 30, 24, 31, 32, 33 | hdmapg 41913 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐺‘((𝐽‘𝑏)‘𝑎)) = ((𝐽‘𝑎)‘𝑏)) |
63 | 34 | fveq2d 6911 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐺‘(𝑎 , 𝑏)) = (𝐺‘((𝐽‘𝑏)‘𝑎))) |
64 | 1, 4, 5, 30, 2, 31, 11, 33, 32 | hlhilipval 41936 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑏 , 𝑎) = ((𝐽‘𝑎)‘𝑏)) |
65 | 62, 63, 64 | 3eqtr4d 2785 | . 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 21690 | 1 ⊢ (𝜑 → 𝑈 ∈ PreHil) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 Basecbs 17245 +gcplusg 17298 .rcmulr 17299 Scalarcsca 17301 ·𝑠 cvsca 17302 ·𝑖cip 17303 0gc0g 17486 LModclmod 20875 PreHilcphl 21660 HLchlt 39332 LHypclh 39967 DVecHcdvh 41061 HDMapchdma 41775 HGMapchg 41866 HLHilchlh 41915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-riotaBAD 38935 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-undef 8297 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-0g 17488 df-mre 17631 df-mrc 17632 df-acs 17634 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18490 df-clat 18557 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-ghm 19244 df-cntz 19348 df-oppg 19377 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-rhm 20489 df-nzr 20530 df-subrg 20587 df-rlreg 20711 df-domn 20712 df-drng 20748 df-staf 20857 df-srng 20858 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lmhm 21039 df-lvec 21120 df-sra 21190 df-rgmod 21191 df-phl 21662 df-lsatoms 38958 df-lshyp 38959 df-lcv 39001 df-lfl 39040 df-lkr 39068 df-ldual 39106 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-llines 39481 df-lplanes 39482 df-lvols 39483 df-lines 39484 df-psubsp 39486 df-pmap 39487 df-padd 39779 df-lhyp 39971 df-laut 39972 df-ldil 40087 df-ltrn 40088 df-trl 40142 df-tgrp 40726 df-tendo 40738 df-edring 40740 df-dveca 40986 df-disoa 41012 df-dvech 41062 df-dib 41122 df-dic 41156 df-dih 41212 df-doch 41331 df-djh 41378 df-lcdual 41570 df-mapd 41608 df-hvmap 41740 df-hdmap1 41776 df-hdmap 41777 df-hgmap 41867 df-hlhil 41916 |
This theorem is referenced by: hlhilhillem 41947 |
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