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| Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap1neglem1N | Structured version Visualization version GIF version | ||
| Description: Lemma for hdmapneg 35939. 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 2609 | . . . . . 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 2609 | . . . . . 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 3551 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
| 21 | 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 18, 19, 15, 20 | hdmap1cl 35895 | . . . . . . 7 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷) |
| 22 | 1, 21 | eqeltrrd 2688 | . . . . . 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 35892 | . . . . 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 220 | . . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋(-g‘𝑈)𝑌)})) = (𝐿‘{(𝐹(-g‘𝐶)𝐺)}))) |
| 25 | 24 | simpld 473 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐿‘{𝐺})) |
| 26 | 2, 3, 14 | dvhlmod 35200 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
| 27 | hdmap1neglem1.r | . . . . . 6 ⊢ 𝑅 = (invg‘𝑈) | |
| 28 | 4, 27, 7 | lspsnneg 18775 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{(𝑅‘𝑌)}) = (𝑁‘{𝑌})) |
| 29 | 26, 20, 28 | syl2anc 690 | . . . 4 ⊢ (𝜑 → (𝑁‘{(𝑅‘𝑌)}) = (𝑁‘{𝑌})) |
| 30 | 29 | fveq2d 6091 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑅‘𝑌)})) = (𝑀‘(𝑁‘{𝑌}))) |
| 31 | 2, 8, 14 | lcdlmod 35682 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
| 32 | hdmap1neglem1.s | . . . . 5 ⊢ 𝑆 = (invg‘𝐶) | |
| 33 | 9, 32, 11 | lspsnneg 18775 | . . . 4 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐷) → (𝐿‘{(𝑆‘𝐺)}) = (𝐿‘{𝐺})) |
| 34 | 31, 22, 33 | syl2anc 690 | . . 3 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝐺)}) = (𝐿‘{𝐺})) |
| 35 | 25, 30, 34 | 3eqtr4d 2653 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑅‘𝑌)})) = (𝐿‘{(𝑆‘𝐺)})) |
| 36 | 24 | simprd 477 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋(-g‘𝑈)𝑌)})) = (𝐿‘{(𝐹(-g‘𝐶)𝐺)})) |
| 37 | lmodabl 18681 | . . . . . . . . 9 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Abel) | |
| 38 | 26, 37 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ Abel) |
| 39 | 15 | eldifad 3551 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
| 40 | 4, 5, 27, 38, 39, 20 | ablsub2inv 17987 | . . . . . . 7 ⊢ (𝜑 → ((𝑅‘𝑋)(-g‘𝑈)(𝑅‘𝑌)) = (𝑌(-g‘𝑈)𝑋)) |
| 41 | 40 | sneqd 4136 | . . . . . 6 ⊢ (𝜑 → {((𝑅‘𝑋)(-g‘𝑈)(𝑅‘𝑌))} = {(𝑌(-g‘𝑈)𝑋)}) |
| 42 | 41 | fveq2d 6091 | . . . . 5 ⊢ (𝜑 → (𝑁‘{((𝑅‘𝑋)(-g‘𝑈)(𝑅‘𝑌))}) = (𝑁‘{(𝑌(-g‘𝑈)𝑋)})) |
| 43 | 4, 5, 7, 26, 20, 39 | lspsnsub 18776 | . . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑌(-g‘𝑈)𝑋)}) = (𝑁‘{(𝑋(-g‘𝑈)𝑌)})) |
| 44 | 42, 43 | eqtrd 2643 | . . . 4 ⊢ (𝜑 → (𝑁‘{((𝑅‘𝑋)(-g‘𝑈)(𝑅‘𝑌))}) = (𝑁‘{(𝑋(-g‘𝑈)𝑌)})) |
| 45 | 44 | fveq2d 6091 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{((𝑅‘𝑋)(-g‘𝑈)(𝑅‘𝑌))})) = (𝑀‘(𝑁‘{(𝑋(-g‘𝑈)𝑌)}))) |
| 46 | lmodabl 18681 | . . . . . . . 8 ⊢ (𝐶 ∈ LMod → 𝐶 ∈ Abel) | |
| 47 | 31, 46 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Abel) |
| 48 | 9, 10, 32, 47, 16, 22 | ablsub2inv 17987 | . . . . . 6 ⊢ (𝜑 → ((𝑆‘𝐹)(-g‘𝐶)(𝑆‘𝐺)) = (𝐺(-g‘𝐶)𝐹)) |
| 49 | 48 | sneqd 4136 | . . . . 5 ⊢ (𝜑 → {((𝑆‘𝐹)(-g‘𝐶)(𝑆‘𝐺))} = {(𝐺(-g‘𝐶)𝐹)}) |
| 50 | 49 | fveq2d 6091 | . . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝐹)(-g‘𝐶)(𝑆‘𝐺))}) = (𝐿‘{(𝐺(-g‘𝐶)𝐹)})) |
| 51 | 9, 10, 11, 31, 22, 16 | lspsnsub 18776 | . . . 4 ⊢ (𝜑 → (𝐿‘{(𝐺(-g‘𝐶)𝐹)}) = (𝐿‘{(𝐹(-g‘𝐶)𝐺)})) |
| 52 | 50, 51 | eqtrd 2643 | . . 3 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝐹)(-g‘𝐶)(𝑆‘𝐺))}) = (𝐿‘{(𝐹(-g‘𝐶)𝐺)})) |
| 53 | 36, 45, 52 | 3eqtr4d 2653 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{((𝑅‘𝑋)(-g‘𝑈)(𝑅‘𝑌))})) = (𝐿‘{((𝑆‘𝐹)(-g‘𝐶)(𝑆‘𝐺))})) |
| 54 | lmodgrp 18641 | . . . . 5 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Grp) | |
| 55 | 26, 54 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ Grp) |
| 56 | 4, 6, 27 | grpinvnzcl 17258 | . . . 4 ⊢ ((𝑈 ∈ Grp ∧ 𝑋 ∈ (𝑉 ∖ { 0 })) → (𝑅‘𝑋) ∈ (𝑉 ∖ { 0 })) |
| 57 | 55, 15, 56 | syl2anc 690 | . . 3 ⊢ (𝜑 → (𝑅‘𝑋) ∈ (𝑉 ∖ { 0 })) |
| 58 | 9, 32 | lmodvnegcl 18675 | . . . 4 ⊢ ((𝐶 ∈ LMod ∧ 𝐹 ∈ 𝐷) → (𝑆‘𝐹) ∈ 𝐷) |
| 59 | 31, 16, 58 | syl2anc 690 | . . 3 ⊢ (𝜑 → (𝑆‘𝐹) ∈ 𝐷) |
| 60 | 4, 6, 27 | grpinvnzcl 17258 | . . . 4 ⊢ ((𝑈 ∈ Grp ∧ 𝑌 ∈ (𝑉 ∖ { 0 })) → (𝑅‘𝑌) ∈ (𝑉 ∖ { 0 })) |
| 61 | 55, 17, 60 | syl2anc 690 | . . 3 ⊢ (𝜑 → (𝑅‘𝑌) ∈ (𝑉 ∖ { 0 })) |
| 62 | 9, 32 | lmodvnegcl 18675 | . . . 4 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐷) → (𝑆‘𝐺) ∈ 𝐷) |
| 63 | 31, 22, 62 | syl2anc 690 | . . 3 ⊢ (𝜑 → (𝑆‘𝐺) ∈ 𝐷) |
| 64 | 4, 27, 7 | lspsnneg 18775 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{(𝑅‘𝑋)}) = (𝑁‘{𝑋})) |
| 65 | 26, 39, 64 | syl2anc 690 | . . . 4 ⊢ (𝜑 → (𝑁‘{(𝑅‘𝑋)}) = (𝑁‘{𝑋})) |
| 66 | 19, 65, 29 | 3netr4d 2858 | . . 3 ⊢ (𝜑 → (𝑁‘{(𝑅‘𝑋)}) ≠ (𝑁‘{(𝑅‘𝑌)})) |
| 67 | 65 | fveq2d 6091 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑅‘𝑋)})) = (𝑀‘(𝑁‘{𝑋}))) |
| 68 | 9, 32, 11 | lspsnneg 18775 | . . . . 5 ⊢ ((𝐶 ∈ LMod ∧ 𝐹 ∈ 𝐷) → (𝐿‘{(𝑆‘𝐹)}) = (𝐿‘{𝐹})) |
| 69 | 31, 16, 68 | syl2anc 690 | . . . 4 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝐹)}) = (𝐿‘{𝐹})) |
| 70 | 18, 67, 69 | 3eqtr4d 2653 | . . 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 35892 | . 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 958 | 1 ⊢ (𝜑 → (𝐼‘⟨(𝑅‘𝑋), (𝑆‘𝐹), (𝑅‘𝑌)⟩) = (𝑆‘𝐺)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 382 = wceq 1474 ∈ wcel 1976 ≠ wne 2779 ∖ cdif 3536 {csn 4124 ⟨cotp 4132 ‘cfv 5789 (class class class)co 6526 Basecbs 15643 0gc0g 15871 Grpcgrp 17193 invgcminusg 17194 -gcsg 17195 Abelcabl 17965 LModclmod 18634 LSpanclspn 18740 HLchlt 33438 LHypclh 34071 DVecHcdvh 35168 LCDualclcd 35676 mapdcmpd 35714 HDMap1chdma1 35882 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1712 ax-4 1727 ax-5 1826 ax-6 1874 ax-7 1921 ax-8 1978 ax-9 1985 ax-10 2005 ax-11 2020 ax-12 2033 ax-13 2233 ax-ext 2589 ax-rep 4693 ax-sep 4703 ax-nul 4711 ax-pow 4763 ax-pr 4827 ax-un 6824 ax-cnex 9848 ax-resscn 9849 ax-1cn 9850 ax-icn 9851 ax-addcl 9852 ax-addrcl 9853 ax-mulcl 9854 ax-mulrcl 9855 ax-mulcom 9856 ax-addass 9857 ax-mulass 9858 ax-distr 9859 ax-i2m1 9860 ax-1ne0 9861 ax-1rid 9862 ax-rnegex 9863 ax-rrecex 9864 ax-cnre 9865 ax-pre-lttri 9866 ax-pre-lttrn 9867 ax-pre-ltadd 9868 ax-pre-mulgt0 9869 ax-riotaBAD 33040 |
| This theorem depends on definitions: df-bi 195 df-or 383 df-an 384 df-3or 1031 df-3an 1032 df-tru 1477 df-fal 1480 df-ex 1695 df-nf 1700 df-sb 1867 df-eu 2461 df-mo 2462 df-clab 2596 df-cleq 2602 df-clel 2605 df-nfc 2739 df-ne 2781 df-nel 2782 df-ral 2900 df-rex 2901 df-reu 2902 df-rmo 2903 df-rab 2904 df-v 3174 df-sbc 3402 df-csb 3499 df-dif 3542 df-un 3544 df-in 3546 df-ss 3553 df-pss 3555 df-nul 3874 df-if 4036 df-pw 4109 df-sn 4125 df-pr 4127 df-tp 4129 df-op 4131 df-ot 4133 df-uni 4367 df-int 4405 df-iun 4451 df-iin 4452 df-br 4578 df-opab 4638 df-mpt 4639 df-tr 4675 df-eprel 4938 df-id 4942 df-po 4948 df-so 4949 df-fr 4986 df-we 4988 df-xp 5033 df-rel 5034 df-cnv 5035 df-co 5036 df-dm 5037 df-rn 5038 df-res 5039 df-ima 5040 df-pred 5582 df-ord 5628 df-on 5629 df-lim 5630 df-suc 5631 df-iota 5753 df-fun 5791 df-fn 5792 df-f 5793 df-f1 5794 df-fo 5795 df-f1o 5796 df-fv 5797 df-riota 6488 df-ov 6529 df-oprab 6530 df-mpt2 6531 df-of 6772 df-om 6935 df-1st 7036 df-2nd 7037 df-tpos 7216 df-undef 7263 df-wrecs 7271 df-recs 7332 df-rdg 7370 df-1o 7424 df-oadd 7428 df-er 7606 df-map 7723 df-en 7819 df-dom 7820 df-sdom 7821 df-fin 7822 df-pnf 9932 df-mnf 9933 df-xr 9934 df-ltxr 9935 df-le 9936 df-sub 10119 df-neg 10120 df-nn 10870 df-2 10928 df-3 10929 df-4 10930 df-5 10931 df-6 10932 df-n0 11142 df-z 11213 df-uz 11522 df-fz 12155 df-struct 15645 df-ndx 15646 df-slot 15647 df-base 15648 df-sets 15649 df-ress 15650 df-plusg 15729 df-mulr 15730 df-sca 15732 df-vsca 15733 df-0g 15873 df-mre 16017 df-mrc 16018 df-acs 16020 df-preset 16699 df-poset 16717 df-plt 16729 df-lub 16745 df-glb 16746 df-join 16747 df-meet 16748 df-p0 16810 df-p1 16811 df-lat 16817 df-clat 16879 df-mgm 17013 df-sgrp 17055 df-mnd 17066 df-submnd 17107 df-grp 17196 df-minusg 17197 df-sbg 17198 df-subg 17362 df-cntz 17521 df-oppg 17547 df-lsm 17822 df-cmn 17966 df-abl 17967 df-mgp 18261 df-ur 18273 df-ring 18320 df-oppr 18394 df-dvdsr 18412 df-unit 18413 df-invr 18443 df-dvr 18454 df-drng 18520 df-lmod 18636 df-lss 18702 df-lsp 18741 df-lvec 18872 df-lsatoms 33064 df-lshyp 33065 df-lcv 33107 df-lfl 33146 df-lkr 33174 df-ldual 33212 df-oposet 33264 df-ol 33266 df-oml 33267 df-covers 33354 df-ats 33355 df-atl 33386 df-cvlat 33410 df-hlat 33439 df-llines 33585 df-lplanes 33586 df-lvols 33587 df-lines 33588 df-psubsp 33590 df-pmap 33591 df-padd 33883 df-lhyp 34075 df-laut 34076 df-ldil 34191 df-ltrn 34192 df-trl 34247 df-tgrp 34832 df-tendo 34844 df-edring 34846 df-dveca 35092 df-disoa 35119 df-dvech 35169 df-dib 35229 df-dic 35263 df-dih 35319 df-doch 35438 df-djh 35485 df-lcdual 35677 df-mapd 35715 df-hdmap1 35884 |
| This theorem is referenced by: (None) |
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