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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlhilphllem | Structured version Visualization version GIF version |
Description: Lemma for hlhil 24195. (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 39573 | . 2 ⊢ (𝜑 → 𝑉 = (Base‘𝑈)) |
7 | hlhilphllem.a | . . 3 ⊢ + = (+g‘𝐿) | |
8 | 1, 2, 3, 4, 7 | hlhilplus 39574 | . 2 ⊢ (𝜑 → + = (+g‘𝑈)) |
9 | hlhilphllem.s | . . 3 ⊢ · = ( ·𝑠 ‘𝐿) | |
10 | 1, 4, 9, 2, 3 | hlhilvsca 39584 | . 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 39592 | . 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 39579 | . 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐹)) |
20 | hlhilphllem.p | . . 3 ⊢ ⨣ = (+g‘𝑅) | |
21 | 1, 4, 17, 2, 15, 3, 20 | hlhilsplus2 39580 | . 2 ⊢ (𝜑 → ⨣ = (+g‘𝐹)) |
22 | hlhilphllem.t | . . 3 ⊢ × = (.r‘𝑅) | |
23 | 1, 4, 17, 2, 15, 3, 22 | hlhilsmul2 39581 | . 2 ⊢ (𝜑 → × = (.r‘𝐹)) |
24 | hlhilphllem.g | . . 3 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) | |
25 | 1, 2, 15, 24, 3 | hlhilnvl 39587 | . 2 ⊢ (𝜑 → 𝐺 = (*𝑟‘𝐹)) |
26 | hlhilphllem.q | . . 3 ⊢ 𝑄 = (0g‘𝑅) | |
27 | 1, 4, 17, 2, 15, 3, 26 | hlhils0 39582 | . 2 ⊢ (𝜑 → 𝑄 = (0g‘𝐹)) |
28 | 1, 2, 3 | hlhillvec 39588 | . 2 ⊢ (𝜑 → 𝑈 ∈ LVec) |
29 | 1, 2, 3, 15 | hlhilsrng 39591 | . 2 ⊢ (𝜑 → 𝐹 ∈ *-Ring) |
30 | hlhilphllem.j | . . . 4 ⊢ 𝐽 = ((HDMap‘𝐾)‘𝑊) | |
31 | 3 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
32 | simp2 1138 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑎 ∈ 𝑉) | |
33 | simp3 1139 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ 𝑉) | |
34 | 1, 4, 5, 30, 2, 31, 11, 32, 33 | hlhilipval 39586 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎 , 𝑏) = ((𝐽‘𝑏)‘𝑎)) |
35 | 1, 4, 5, 17, 18, 30, 31, 32, 33 | hdmapipcl 39542 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝐽‘𝑏)‘𝑎) ∈ 𝐵) |
36 | 34, 35 | eqeltrd 2833 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎 , 𝑏) ∈ 𝐵) |
37 | 3 | 3ad2ant1 1134 | . . . 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 1138 | . . . 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 39543 | . . 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 38747 | . . . . . 6 ⊢ (𝜑 → 𝐿 ∈ LMod) |
44 | 43 | 3ad2ant1 1134 | . . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝐿 ∈ LMod) |
45 | 5, 17, 9, 18 | lmodvscl 19770 | . . . . . 6 ⊢ ((𝐿 ∈ LMod ∧ 𝑑 ∈ 𝐵 ∧ 𝑎 ∈ 𝑉) → (𝑑 · 𝑎) ∈ 𝑉) |
46 | 44, 41, 38, 45 | syl3anc 1372 | . . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑑 · 𝑎) ∈ 𝑉) |
47 | 5, 7 | lmodvacl 19767 | . . . . 5 ⊢ ((𝐿 ∈ LMod ∧ (𝑑 · 𝑎) ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝑑 · 𝑎) + 𝑏) ∈ 𝑉) |
48 | 44, 46, 39, 47 | syl3anc 1372 | . . . 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 39586 | . . 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 39586 | . . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑎 , 𝑐) = ((𝐽‘𝑐)‘𝑎)) |
51 | 50 | oveq2d 7186 | . . . 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 39586 | . . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑏 , 𝑐) = ((𝐽‘𝑐)‘𝑏)) |
53 | 51, 52 | oveq12d 7188 | . . 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 2783 | . 2 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (((𝑑 · 𝑎) + 𝑏) , 𝑐) = ((𝑑 × (𝑎 , 𝑐)) ⨣ (𝑏 , 𝑐))) |
55 | 3 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
56 | simpr 488 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → 𝑎 ∈ 𝑉) | |
57 | 1, 4, 5, 30, 2, 55, 11, 56, 56 | hlhilipval 39586 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → (𝑎 , 𝑎) = ((𝐽‘𝑎)‘𝑎)) |
58 | 57 | eqeq1d 2740 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → ((𝑎 , 𝑎) = 𝑄 ↔ ((𝐽‘𝑎)‘𝑎) = 𝑄)) |
59 | 1, 4, 5, 13, 17, 26, 30, 55, 56 | hdmapip0 39552 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → (((𝐽‘𝑎)‘𝑎) = 𝑄 ↔ 𝑎 = 0 )) |
60 | 58, 59 | bitrd 282 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉) → ((𝑎 , 𝑎) = 𝑄 ↔ 𝑎 = 0 )) |
61 | 60 | biimp3a 1470 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ (𝑎 , 𝑎) = 𝑄) → 𝑎 = 0 ) |
62 | 1, 4, 5, 30, 24, 31, 32, 33 | hdmapg 39567 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐺‘((𝐽‘𝑏)‘𝑎)) = ((𝐽‘𝑎)‘𝑏)) |
63 | 34 | fveq2d 6678 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐺‘(𝑎 , 𝑏)) = (𝐺‘((𝐽‘𝑏)‘𝑎))) |
64 | 1, 4, 5, 30, 2, 31, 11, 33, 32 | hlhilipval 39586 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑏 , 𝑎) = ((𝐽‘𝑎)‘𝑏)) |
65 | 62, 63, 64 | 3eqtr4d 2783 | . 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 20470 | 1 ⊢ (𝜑 → 𝑈 ∈ PreHil) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ‘cfv 6339 (class class class)co 7170 ∈ cmpo 7172 Basecbs 16586 +gcplusg 16668 .rcmulr 16669 Scalarcsca 16671 ·𝑠 cvsca 16672 ·𝑖cip 16673 0gc0g 16816 LModclmod 19753 PreHilcphl 20440 HLchlt 36987 LHypclh 37621 DVecHcdvh 38715 HDMapchdma 39429 HGMapchg 39520 HLHilchlh 39569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-riotaBAD 36590 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-ot 4525 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-om 7600 df-1st 7714 df-2nd 7715 df-tpos 7921 df-undef 7968 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-n0 11977 df-z 12063 df-uz 12325 df-fz 12982 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-starv 16683 df-sca 16684 df-vsca 16685 df-ip 16686 df-0g 16818 df-mre 16960 df-mrc 16961 df-acs 16963 df-proset 17654 df-poset 17672 df-plt 17684 df-lub 17700 df-glb 17701 df-join 17702 df-meet 17703 df-p0 17765 df-p1 17766 df-lat 17772 df-clat 17834 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-mhm 18072 df-submnd 18073 df-grp 18222 df-minusg 18223 df-sbg 18224 df-subg 18394 df-ghm 18474 df-cntz 18565 df-oppg 18592 df-lsm 18879 df-cmn 19026 df-abl 19027 df-mgp 19359 df-ur 19371 df-ring 19418 df-oppr 19495 df-dvdsr 19513 df-unit 19514 df-invr 19544 df-dvr 19555 df-rnghom 19589 df-drng 19623 df-subrg 19652 df-staf 19735 df-srng 19736 df-lmod 19755 df-lss 19823 df-lsp 19863 df-lmhm 19913 df-lvec 19994 df-sra 20063 df-rgmod 20064 df-phl 20442 df-lsatoms 36613 df-lshyp 36614 df-lcv 36656 df-lfl 36695 df-lkr 36723 df-ldual 36761 df-oposet 36813 df-ol 36815 df-oml 36816 df-covers 36903 df-ats 36904 df-atl 36935 df-cvlat 36959 df-hlat 36988 df-llines 37135 df-lplanes 37136 df-lvols 37137 df-lines 37138 df-psubsp 37140 df-pmap 37141 df-padd 37433 df-lhyp 37625 df-laut 37626 df-ldil 37741 df-ltrn 37742 df-trl 37796 df-tgrp 38380 df-tendo 38392 df-edring 38394 df-dveca 38640 df-disoa 38666 df-dvech 38716 df-dib 38776 df-dic 38810 df-dih 38866 df-doch 38985 df-djh 39032 df-lcdual 39224 df-mapd 39262 df-hvmap 39394 df-hdmap1 39430 df-hdmap 39431 df-hgmap 39521 df-hlhil 39570 |
This theorem is referenced by: hlhilhillem 39597 |
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