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| Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap1neglem1N | Structured version Visualization version GIF version | ||
| Description: Lemma for hdmapneg 37455. TODO: Not used; delete. (Contributed by NM, 23-May-2015.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| hdmap1neglem1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
| hdmap1neglem1.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
| hdmap1neglem1.v | ⊢ 𝑉 = (Base‘𝑈) |
| hdmap1neglem1.r | ⊢ 𝑅 = (invg‘𝑈) |
| hdmap1neglem1.o | ⊢ 0 = (0g‘𝑈) |
| hdmap1neglem1.n | ⊢ 𝑁 = (LSpan‘𝑈) |
| hdmap1neglem1.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
| hdmap1neglem1.d | ⊢ 𝐷 = (Base‘𝐶) |
| hdmap1neglem1.s | ⊢ 𝑆 = (invg‘𝐶) |
| hdmap1neglem1.l | ⊢ 𝐿 = (LSpan‘𝐶) |
| hdmap1neglem1.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
| hdmap1neglem1.i | ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) |
| hdmap1neglem1.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
| hdmap1neglem1.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
| hdmap1neglem1.mn | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹})) |
| hdmap1neglem1.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
| hdmap1neglem1.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
| hdmap1neglem1.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
| hdmap1neglem1.e | ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺) |
| Ref | Expression |
|---|---|
| hdmap1neglem1N | ⊢ (𝜑 → (𝐼‘⟨(𝑅‘𝑋), (𝑆‘𝐹), (𝑅‘𝑌)⟩) = (𝑆‘𝐺)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hdmap1neglem1.e | . . . . 5 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺) | |
| 2 | hdmap1neglem1.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 3 | hdmap1neglem1.u | . . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
| 4 | hdmap1neglem1.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑈) | |
| 5 | eqid 2651 | . . . . . 6 ⊢ (-g‘𝑈) = (-g‘𝑈) | |
| 6 | hdmap1neglem1.o | . . . . . 6 ⊢ 0 = (0g‘𝑈) | |
| 7 | hdmap1neglem1.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈) | |
| 8 | hdmap1neglem1.c | . . . . . 6 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
| 9 | hdmap1neglem1.d | . . . . . 6 ⊢ 𝐷 = (Base‘𝐶) | |
| 10 | eqid 2651 | . . . . . 6 ⊢ (-g‘𝐶) = (-g‘𝐶) | |
| 11 | hdmap1neglem1.l | . . . . . 6 ⊢ 𝐿 = (LSpan‘𝐶) | |
| 12 | hdmap1neglem1.m | . . . . . 6 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
| 13 | hdmap1neglem1.i | . . . . . 6 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) | |
| 14 | hdmap1neglem1.k | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
| 15 | hdmap1neglem1.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
| 16 | hdmap1neglem1.f | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
| 17 | hdmap1neglem1.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
| 18 | hdmap1neglem1.mn | . . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹})) | |
| 19 | hdmap1neglem1.ne | . . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
| 20 | 17 | eldifad 3619 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
| 21 | 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 18, 19, 15, 20 | hdmap1cl 37411 | . . . . . . 7 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷) |
| 22 | 1, 21 | eqeltrrd 2731 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐷) |
| 23 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 22, 19, 18 | hdmap1eq 37408 | . . . . 5 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋(-g‘𝑈)𝑌)})) = (𝐿‘{(𝐹(-g‘𝐶)𝐺)})))) |
| 24 | 1, 23 | mpbid 222 | . . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋(-g‘𝑈)𝑌)})) = (𝐿‘{(𝐹(-g‘𝐶)𝐺)}))) |
| 25 | 24 | simpld 474 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐿‘{𝐺})) |
| 26 | 2, 3, 14 | dvhlmod 36716 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
| 27 | hdmap1neglem1.r | . . . . . 6 ⊢ 𝑅 = (invg‘𝑈) | |
| 28 | 4, 27, 7 | lspsnneg 19054 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{(𝑅‘𝑌)}) = (𝑁‘{𝑌})) |
| 29 | 26, 20, 28 | syl2anc 694 | . . . 4 ⊢ (𝜑 → (𝑁‘{(𝑅‘𝑌)}) = (𝑁‘{𝑌})) |
| 30 | 29 | fveq2d 6233 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑅‘𝑌)})) = (𝑀‘(𝑁‘{𝑌}))) |
| 31 | 2, 8, 14 | lcdlmod 37198 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
| 32 | hdmap1neglem1.s | . . . . 5 ⊢ 𝑆 = (invg‘𝐶) | |
| 33 | 9, 32, 11 | lspsnneg 19054 | . . . 4 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐷) → (𝐿‘{(𝑆‘𝐺)}) = (𝐿‘{𝐺})) |
| 34 | 31, 22, 33 | syl2anc 694 | . . 3 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝐺)}) = (𝐿‘{𝐺})) |
| 35 | 25, 30, 34 | 3eqtr4d 2695 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑅‘𝑌)})) = (𝐿‘{(𝑆‘𝐺)})) |
| 36 | 24 | simprd 478 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋(-g‘𝑈)𝑌)})) = (𝐿‘{(𝐹(-g‘𝐶)𝐺)})) |
| 37 | lmodabl 18958 | . . . . . . . . 9 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Abel) | |
| 38 | 26, 37 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ Abel) |
| 39 | 15 | eldifad 3619 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
| 40 | 4, 5, 27, 38, 39, 20 | ablsub2inv 18262 | . . . . . . 7 ⊢ (𝜑 → ((𝑅‘𝑋)(-g‘𝑈)(𝑅‘𝑌)) = (𝑌(-g‘𝑈)𝑋)) |
| 41 | 40 | sneqd 4222 | . . . . . 6 ⊢ (𝜑 → {((𝑅‘𝑋)(-g‘𝑈)(𝑅‘𝑌))} = {(𝑌(-g‘𝑈)𝑋)}) |
| 42 | 41 | fveq2d 6233 | . . . . 5 ⊢ (𝜑 → (𝑁‘{((𝑅‘𝑋)(-g‘𝑈)(𝑅‘𝑌))}) = (𝑁‘{(𝑌(-g‘𝑈)𝑋)})) |
| 43 | 4, 5, 7, 26, 20, 39 | lspsnsub 19055 | . . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑌(-g‘𝑈)𝑋)}) = (𝑁‘{(𝑋(-g‘𝑈)𝑌)})) |
| 44 | 42, 43 | eqtrd 2685 | . . . 4 ⊢ (𝜑 → (𝑁‘{((𝑅‘𝑋)(-g‘𝑈)(𝑅‘𝑌))}) = (𝑁‘{(𝑋(-g‘𝑈)𝑌)})) |
| 45 | 44 | fveq2d 6233 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{((𝑅‘𝑋)(-g‘𝑈)(𝑅‘𝑌))})) = (𝑀‘(𝑁‘{(𝑋(-g‘𝑈)𝑌)}))) |
| 46 | lmodabl 18958 | . . . . . . . 8 ⊢ (𝐶 ∈ LMod → 𝐶 ∈ Abel) | |
| 47 | 31, 46 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Abel) |
| 48 | 9, 10, 32, 47, 16, 22 | ablsub2inv 18262 | . . . . . 6 ⊢ (𝜑 → ((𝑆‘𝐹)(-g‘𝐶)(𝑆‘𝐺)) = (𝐺(-g‘𝐶)𝐹)) |
| 49 | 48 | sneqd 4222 | . . . . 5 ⊢ (𝜑 → {((𝑆‘𝐹)(-g‘𝐶)(𝑆‘𝐺))} = {(𝐺(-g‘𝐶)𝐹)}) |
| 50 | 49 | fveq2d 6233 | . . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝐹)(-g‘𝐶)(𝑆‘𝐺))}) = (𝐿‘{(𝐺(-g‘𝐶)𝐹)})) |
| 51 | 9, 10, 11, 31, 22, 16 | lspsnsub 19055 | . . . 4 ⊢ (𝜑 → (𝐿‘{(𝐺(-g‘𝐶)𝐹)}) = (𝐿‘{(𝐹(-g‘𝐶)𝐺)})) |
| 52 | 50, 51 | eqtrd 2685 | . . 3 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝐹)(-g‘𝐶)(𝑆‘𝐺))}) = (𝐿‘{(𝐹(-g‘𝐶)𝐺)})) |
| 53 | 36, 45, 52 | 3eqtr4d 2695 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{((𝑅‘𝑋)(-g‘𝑈)(𝑅‘𝑌))})) = (𝐿‘{((𝑆‘𝐹)(-g‘𝐶)(𝑆‘𝐺))})) |
| 54 | lmodgrp 18918 | . . . . 5 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Grp) | |
| 55 | 26, 54 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ Grp) |
| 56 | 4, 6, 27 | grpinvnzcl 17534 | . . . 4 ⊢ ((𝑈 ∈ Grp ∧ 𝑋 ∈ (𝑉 ∖ { 0 })) → (𝑅‘𝑋) ∈ (𝑉 ∖ { 0 })) |
| 57 | 55, 15, 56 | syl2anc 694 | . . 3 ⊢ (𝜑 → (𝑅‘𝑋) ∈ (𝑉 ∖ { 0 })) |
| 58 | 9, 32 | lmodvnegcl 18952 | . . . 4 ⊢ ((𝐶 ∈ LMod ∧ 𝐹 ∈ 𝐷) → (𝑆‘𝐹) ∈ 𝐷) |
| 59 | 31, 16, 58 | syl2anc 694 | . . 3 ⊢ (𝜑 → (𝑆‘𝐹) ∈ 𝐷) |
| 60 | 4, 6, 27 | grpinvnzcl 17534 | . . . 4 ⊢ ((𝑈 ∈ Grp ∧ 𝑌 ∈ (𝑉 ∖ { 0 })) → (𝑅‘𝑌) ∈ (𝑉 ∖ { 0 })) |
| 61 | 55, 17, 60 | syl2anc 694 | . . 3 ⊢ (𝜑 → (𝑅‘𝑌) ∈ (𝑉 ∖ { 0 })) |
| 62 | 9, 32 | lmodvnegcl 18952 | . . . 4 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐷) → (𝑆‘𝐺) ∈ 𝐷) |
| 63 | 31, 22, 62 | syl2anc 694 | . . 3 ⊢ (𝜑 → (𝑆‘𝐺) ∈ 𝐷) |
| 64 | 4, 27, 7 | lspsnneg 19054 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{(𝑅‘𝑋)}) = (𝑁‘{𝑋})) |
| 65 | 26, 39, 64 | syl2anc 694 | . . . 4 ⊢ (𝜑 → (𝑁‘{(𝑅‘𝑋)}) = (𝑁‘{𝑋})) |
| 66 | 19, 65, 29 | 3netr4d 2900 | . . 3 ⊢ (𝜑 → (𝑁‘{(𝑅‘𝑋)}) ≠ (𝑁‘{(𝑅‘𝑌)})) |
| 67 | 65 | fveq2d 6233 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑅‘𝑋)})) = (𝑀‘(𝑁‘{𝑋}))) |
| 68 | 9, 32, 11 | lspsnneg 19054 | . . . . 5 ⊢ ((𝐶 ∈ LMod ∧ 𝐹 ∈ 𝐷) → (𝐿‘{(𝑆‘𝐹)}) = (𝐿‘{𝐹})) |
| 69 | 31, 16, 68 | syl2anc 694 | . . . 4 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝐹)}) = (𝐿‘{𝐹})) |
| 70 | 18, 67, 69 | 3eqtr4d 2695 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑅‘𝑋)})) = (𝐿‘{(𝑆‘𝐹)})) |
| 71 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 57, 59, 61, 63, 66, 70 | hdmap1eq 37408 | . 2 ⊢ (𝜑 → ((𝐼‘⟨(𝑅‘𝑋), (𝑆‘𝐹), (𝑅‘𝑌)⟩) = (𝑆‘𝐺) ↔ ((𝑀‘(𝑁‘{(𝑅‘𝑌)})) = (𝐿‘{(𝑆‘𝐺)}) ∧ (𝑀‘(𝑁‘{((𝑅‘𝑋)(-g‘𝑈)(𝑅‘𝑌))})) = (𝐿‘{((𝑆‘𝐹)(-g‘𝐶)(𝑆‘𝐺))})))) |
| 72 | 35, 53, 71 | mpbir2and 977 | 1 ⊢ (𝜑 → (𝐼‘⟨(𝑅‘𝑋), (𝑆‘𝐹), (𝑅‘𝑌)⟩) = (𝑆‘𝐺)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ∖ cdif 3604 {csn 4210 ⟨cotp 4218 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 0gc0g 16147 Grpcgrp 17469 invgcminusg 17470 -gcsg 17471 Abelcabl 18240 LModclmod 18911 LSpanclspn 19019 HLchlt 34955 LHypclh 35588 DVecHcdvh 36684 LCDualclcd 37192 mapdcmpd 37230 HDMap1chdma1 37398 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-riotaBAD 34557 |
| This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-ot 4219 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-tpos 7397 df-undef 7444 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-sca 16004 df-vsca 16005 df-0g 16149 df-mre 16293 df-mrc 16294 df-acs 16296 df-preset 16975 df-poset 16993 df-plt 17005 df-lub 17021 df-glb 17022 df-join 17023 df-meet 17024 df-p0 17086 df-p1 17087 df-lat 17093 df-clat 17155 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-grp 17472 df-minusg 17473 df-sbg 17474 df-subg 17638 df-cntz 17796 df-oppg 17822 df-lsm 18097 df-cmn 18241 df-abl 18242 df-mgp 18536 df-ur 18548 df-ring 18595 df-oppr 18669 df-dvdsr 18687 df-unit 18688 df-invr 18718 df-dvr 18729 df-drng 18797 df-lmod 18913 df-lss 18981 df-lsp 19020 df-lvec 19151 df-lsatoms 34581 df-lshyp 34582 df-lcv 34624 df-lfl 34663 df-lkr 34691 df-ldual 34729 df-oposet 34781 df-ol 34783 df-oml 34784 df-covers 34871 df-ats 34872 df-atl 34903 df-cvlat 34927 df-hlat 34956 df-llines 35102 df-lplanes 35103 df-lvols 35104 df-lines 35105 df-psubsp 35107 df-pmap 35108 df-padd 35400 df-lhyp 35592 df-laut 35593 df-ldil 35708 df-ltrn 35709 df-trl 35764 df-tgrp 36348 df-tendo 36360 df-edring 36362 df-dveca 36608 df-disoa 36635 df-dvech 36685 df-dib 36745 df-dic 36779 df-dih 36835 df-doch 36954 df-djh 37001 df-lcdual 37193 df-mapd 37231 df-hdmap1 37400 |
| This theorem is referenced by: (None) |
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