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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlhilphllem | Structured version Visualization version GIF version |
Description: Lemma for hlhil 23973. (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 38952 | . 2 ⊢ (𝜑 → 𝑉 = (Base‘𝑈)) |
7 | hlhilphllem.a | . . 3 ⊢ + = (+g‘𝐿) | |
8 | 1, 2, 3, 4, 7 | hlhilplus 38953 | . 2 ⊢ (𝜑 → + = (+g‘𝑈)) |
9 | hlhilphllem.s | . . 3 ⊢ · = ( ·𝑠 ‘𝐿) | |
10 | 1, 4, 9, 2, 3 | hlhilvsca 38963 | . 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 38971 | . 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 38958 | . 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐹)) |
20 | hlhilphllem.p | . . 3 ⊢ ⨣ = (+g‘𝑅) | |
21 | 1, 4, 17, 2, 15, 3, 20 | hlhilsplus2 38959 | . 2 ⊢ (𝜑 → ⨣ = (+g‘𝐹)) |
22 | hlhilphllem.t | . . 3 ⊢ × = (.r‘𝑅) | |
23 | 1, 4, 17, 2, 15, 3, 22 | hlhilsmul2 38960 | . 2 ⊢ (𝜑 → × = (.r‘𝐹)) |
24 | hlhilphllem.g | . . 3 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) | |
25 | 1, 2, 15, 24, 3 | hlhilnvl 38966 | . 2 ⊢ (𝜑 → 𝐺 = (*𝑟‘𝐹)) |
26 | hlhilphllem.q | . . 3 ⊢ 𝑄 = (0g‘𝑅) | |
27 | 1, 4, 17, 2, 15, 3, 26 | hlhils0 38961 | . 2 ⊢ (𝜑 → 𝑄 = (0g‘𝐹)) |
28 | 1, 2, 3 | hlhillvec 38967 | . 2 ⊢ (𝜑 → 𝑈 ∈ LVec) |
29 | 1, 2, 3, 15 | hlhilsrng 38970 | . 2 ⊢ (𝜑 → 𝐹 ∈ *-Ring) |
30 | hlhilphllem.j | . . . 4 ⊢ 𝐽 = ((HDMap‘𝐾)‘𝑊) | |
31 | 3 | 3ad2ant1 1125 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
32 | simp2 1129 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑎 ∈ 𝑉) | |
33 | simp3 1130 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ 𝑉) | |
34 | 1, 4, 5, 30, 2, 31, 11, 32, 33 | hlhilipval 38965 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎 , 𝑏) = ((𝐽‘𝑏)‘𝑎)) |
35 | 1, 4, 5, 17, 18, 30, 31, 32, 33 | hdmapipcl 38921 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝐽‘𝑏)‘𝑎) ∈ 𝐵) |
36 | 34, 35 | eqeltrd 2910 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎 , 𝑏) ∈ 𝐵) |
37 | 3 | 3ad2ant1 1125 | . . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
38 | simp31 1201 | . . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝑎 ∈ 𝑉) | |
39 | simp32 1202 | . . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝑏 ∈ 𝑉) | |
40 | simp33 1203 | . . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝑐 ∈ 𝑉) | |
41 | simp2 1129 | . . . 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 38922 | . . 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 38126 | . . . . . 6 ⊢ (𝜑 → 𝐿 ∈ LMod) |
44 | 43 | 3ad2ant1 1125 | . . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝐿 ∈ LMod) |
45 | 5, 17, 9, 18 | lmodvscl 19580 | . . . . . 6 ⊢ ((𝐿 ∈ LMod ∧ 𝑑 ∈ 𝐵 ∧ 𝑎 ∈ 𝑉) → (𝑑 · 𝑎) ∈ 𝑉) |
46 | 44, 41, 38, 45 | syl3anc 1363 | . . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑑 · 𝑎) ∈ 𝑉) |
47 | 5, 7 | lmodvacl 19577 | . . . . 5 ⊢ ((𝐿 ∈ LMod ∧ (𝑑 · 𝑎) ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝑑 · 𝑎) + 𝑏) ∈ 𝑉) |
48 | 44, 46, 39, 47 | syl3anc 1363 | . . . 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 38965 | . . 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 38965 | . . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑎 , 𝑐) = ((𝐽‘𝑐)‘𝑎)) |
51 | 50 | oveq2d 7161 | . . . 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 38965 | . . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑏 , 𝑐) = ((𝐽‘𝑐)‘𝑏)) |
53 | 51, 52 | oveq12d 7163 | . . 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 2863 | . 2 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (((𝑑 · 𝑎) + 𝑏) , 𝑐) = ((𝑑 × (𝑎 , 𝑐)) ⨣ (𝑏 , 𝑐))) |
55 | 3 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
56 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → 𝑎 ∈ 𝑉) | |
57 | 1, 4, 5, 30, 2, 55, 11, 56, 56 | hlhilipval 38965 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → (𝑎 , 𝑎) = ((𝐽‘𝑎)‘𝑎)) |
58 | 57 | eqeq1d 2820 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → ((𝑎 , 𝑎) = 𝑄 ↔ ((𝐽‘𝑎)‘𝑎) = 𝑄)) |
59 | 1, 4, 5, 13, 17, 26, 30, 55, 56 | hdmapip0 38931 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → (((𝐽‘𝑎)‘𝑎) = 𝑄 ↔ 𝑎 = 0 )) |
60 | 58, 59 | bitrd 280 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → ((𝑎 , 𝑎) = 𝑄 ↔ 𝑎 = 0 )) |
61 | 60 | biimp3a 1460 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ (𝑎 , 𝑎) = 𝑄) → 𝑎 = 0 ) |
62 | 1, 4, 5, 30, 24, 31, 32, 33 | hdmapg 38946 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐺‘((𝐽‘𝑏)‘𝑎)) = ((𝐽‘𝑎)‘𝑏)) |
63 | 34 | fveq2d 6667 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐺‘(𝑎 , 𝑏)) = (𝐺‘((𝐽‘𝑏)‘𝑎))) |
64 | 1, 4, 5, 30, 2, 31, 11, 33, 32 | hlhilipval 38965 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑏 , 𝑎) = ((𝐽‘𝑎)‘𝑏)) |
65 | 62, 63, 64 | 3eqtr4d 2863 | . 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 20726 | 1 ⊢ (𝜑 → 𝑈 ∈ PreHil) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 Basecbs 16471 +gcplusg 16553 .rcmulr 16554 Scalarcsca 16556 ·𝑠 cvsca 16557 ·𝑖cip 16558 0gc0g 16701 LModclmod 19563 PreHilcphl 20696 HLchlt 36366 LHypclh 37000 DVecHcdvh 38094 HDMapchdma 38808 HGMapchg 38899 HLHilchlh 38948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-riotaBAD 35969 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-ot 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-undef 7928 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-0g 16703 df-mre 16845 df-mrc 16846 df-acs 16848 df-proset 17526 df-poset 17544 df-plt 17556 df-lub 17572 df-glb 17573 df-join 17574 df-meet 17575 df-p0 17637 df-p1 17638 df-lat 17644 df-clat 17706 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-ghm 18294 df-cntz 18385 df-oppg 18412 df-lsm 18690 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-rnghom 19396 df-drng 19433 df-subrg 19462 df-staf 19545 df-srng 19546 df-lmod 19565 df-lss 19633 df-lsp 19673 df-lmhm 19723 df-lvec 19804 df-sra 19873 df-rgmod 19874 df-phl 20698 df-lsatoms 35992 df-lshyp 35993 df-lcv 36035 df-lfl 36074 df-lkr 36102 df-ldual 36140 df-oposet 36192 df-ol 36194 df-oml 36195 df-covers 36282 df-ats 36283 df-atl 36314 df-cvlat 36338 df-hlat 36367 df-llines 36514 df-lplanes 36515 df-lvols 36516 df-lines 36517 df-psubsp 36519 df-pmap 36520 df-padd 36812 df-lhyp 37004 df-laut 37005 df-ldil 37120 df-ltrn 37121 df-trl 37175 df-tgrp 37759 df-tendo 37771 df-edring 37773 df-dveca 38019 df-disoa 38045 df-dvech 38095 df-dib 38155 df-dic 38189 df-dih 38245 df-doch 38364 df-djh 38411 df-lcdual 38603 df-mapd 38641 df-hvmap 38773 df-hdmap1 38809 df-hdmap 38810 df-hgmap 38900 df-hlhil 38949 |
This theorem is referenced by: hlhilhillem 38976 |
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