| Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap11lem1 | Structured version Visualization version GIF version | ||
| Description: Lemma for hdmapadd 41845. (Contributed by NM, 26-May-2015.) |
| Ref | Expression |
|---|---|
| hdmap11.h | ⊢ 𝐻 = (LHyp‘𝐾) |
| hdmap11.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
| hdmap11.v | ⊢ 𝑉 = (Base‘𝑈) |
| hdmap11.p | ⊢ + = (+g‘𝑈) |
| hdmap11.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
| hdmap11.a | ⊢ ✚ = (+g‘𝐶) |
| hdmap11.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
| hdmap11.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
| hdmap11.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
| hdmap11.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
| hdmap11.e | ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 |
| hdmap11.o | ⊢ 0 = (0g‘𝑈) |
| hdmap11.n | ⊢ 𝑁 = (LSpan‘𝑈) |
| hdmap11.d | ⊢ 𝐷 = (Base‘𝐶) |
| hdmap11.l | ⊢ 𝐿 = (LSpan‘𝐶) |
| hdmap11.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
| hdmap11.j | ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) |
| hdmap11.i | ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) |
| hdmap11lem0.1a | ⊢ (𝜑 → 𝑧 ∈ 𝑉) |
| hdmap11lem0.6 | ⊢ (𝜑 → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌})) |
| hdmap11lem0.2a | ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{𝐸})) |
| Ref | Expression |
|---|---|
| hdmap11lem1 | ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) ✚ (𝑆‘𝑌))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hdmap11.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 2 | hdmap11.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
| 3 | hdmap11.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
| 4 | hdmap11.p | . . 3 ⊢ + = (+g‘𝑈) | |
| 5 | hdmap11.o | . . 3 ⊢ 0 = (0g‘𝑈) | |
| 6 | hdmap11.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
| 7 | hdmap11.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
| 8 | hdmap11.d | . . 3 ⊢ 𝐷 = (Base‘𝐶) | |
| 9 | hdmap11.a | . . 3 ⊢ ✚ = (+g‘𝐶) | |
| 10 | hdmap11.l | . . 3 ⊢ 𝐿 = (LSpan‘𝐶) | |
| 11 | hdmap11.m | . . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
| 12 | hdmap11.i | . . 3 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) | |
| 13 | hdmap11.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
| 14 | eqid 2737 | . . . . . 6 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
| 15 | hdmap11.j | . . . . . 6 ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) | |
| 16 | eqid 2737 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 17 | eqid 2737 | . . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
| 18 | hdmap11.e | . . . . . . 7 ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 | |
| 19 | 1, 16, 17, 2, 3, 5, 18, 13 | dvheveccl 41114 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ { 0 })) |
| 20 | 1, 2, 3, 5, 7, 8, 14, 15, 13, 19 | hvmapcl2 41768 | . . . . 5 ⊢ (𝜑 → (𝐽‘𝐸) ∈ (𝐷 ∖ {(0g‘𝐶)})) |
| 21 | 20 | eldifad 3963 | . . . 4 ⊢ (𝜑 → (𝐽‘𝐸) ∈ 𝐷) |
| 22 | 1, 2, 3, 5, 6, 7, 10, 11, 15, 13, 19 | mapdhvmap 41771 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝐸})) = (𝐿‘{(𝐽‘𝐸)})) |
| 23 | hdmap11lem0.2a | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{𝐸})) | |
| 24 | 23 | necomd 2996 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝐸}) ≠ (𝑁‘{𝑧})) |
| 25 | hdmap11lem0.1a | . . . 4 ⊢ (𝜑 → 𝑧 ∈ 𝑉) | |
| 26 | 1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 21, 22, 24, 19, 25 | hdmap1cl 41806 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉) ∈ 𝐷) |
| 27 | eqid 2737 | . . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
| 28 | 1, 2, 13 | dvhlmod 41112 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
| 29 | hdmap11.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
| 30 | hdmap11.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
| 31 | 3, 27, 6, 28, 29, 30 | lspprcl 20976 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈)) |
| 32 | hdmap11lem0.6 | . . . 4 ⊢ (𝜑 → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌})) | |
| 33 | 5, 27, 28, 31, 25, 32 | lssneln0 20951 | . . 3 ⊢ (𝜑 → 𝑧 ∈ (𝑉 ∖ { 0 })) |
| 34 | eqidd 2738 | . . . . 5 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉)) | |
| 35 | eqid 2737 | . . . . . 6 ⊢ (-g‘𝑈) = (-g‘𝑈) | |
| 36 | eqid 2737 | . . . . . 6 ⊢ (-g‘𝐶) = (-g‘𝐶) | |
| 37 | 1, 2, 3, 35, 5, 6, 7, 8, 36, 10, 11, 12, 13, 19, 21, 33, 26, 24, 22 | hdmap1eq 41803 | . . . . 5 ⊢ (𝜑 → ((𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉) ↔ ((𝑀‘(𝑁‘{𝑧})) = (𝐿‘{(𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉)}) ∧ (𝑀‘(𝑁‘{(𝐸(-g‘𝑈)𝑧)})) = (𝐿‘{((𝐽‘𝐸)(-g‘𝐶)(𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉))})))) |
| 38 | 34, 37 | mpbid 232 | . . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑧})) = (𝐿‘{(𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉)}) ∧ (𝑀‘(𝑁‘{(𝐸(-g‘𝑈)𝑧)})) = (𝐿‘{((𝐽‘𝐸)(-g‘𝐶)(𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉))}))) |
| 39 | 38 | simpld 494 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑧})) = (𝐿‘{(𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉)})) |
| 40 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 26, 33, 29, 30, 32, 39 | hdmap1l6 41823 | . 2 ⊢ (𝜑 → (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), (𝑋 + 𝑌)〉) = ((𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑋〉) ✚ (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑌〉))) |
| 41 | hdmap11.s | . . 3 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
| 42 | 3, 4 | lmodvacl 20873 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉) |
| 43 | 28, 29, 30, 42 | syl3anc 1373 | . . 3 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉) |
| 44 | 1, 2, 13 | dvhlvec 41111 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LVec) |
| 45 | 19 | eldifad 3963 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑉) |
| 46 | 3, 4, 6, 28, 29, 30 | lspprvacl 20997 | . . . . . . 7 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝑁‘{𝑋, 𝑌})) |
| 47 | 27, 6, 28, 31, 46 | ellspsn5 20994 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ⊆ (𝑁‘{𝑋, 𝑌})) |
| 48 | 47, 32 | ssneldd 3986 | . . . . 5 ⊢ (𝜑 → ¬ 𝑧 ∈ (𝑁‘{(𝑋 + 𝑌)})) |
| 49 | 3, 6, 28, 25, 43, 48 | lspsnne2 21120 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{(𝑋 + 𝑌)})) |
| 50 | 3, 6, 5, 44, 45, 43, 33, 23, 49 | hdmaplem4 41776 | . . 3 ⊢ (𝜑 → ¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{(𝑋 + 𝑌)}))) |
| 51 | 1, 18, 2, 3, 6, 7, 8, 15, 12, 41, 13, 43, 25, 50 | hdmapval2 41834 | . 2 ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), (𝑋 + 𝑌)〉)) |
| 52 | 3, 6, 44, 25, 29, 30, 32 | lspindpi 21134 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑌}))) |
| 53 | 52 | simpld 494 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋})) |
| 54 | 3, 6, 5, 44, 45, 29, 33, 23, 53 | hdmaplem4 41776 | . . . 4 ⊢ (𝜑 → ¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑋}))) |
| 55 | 1, 18, 2, 3, 6, 7, 8, 15, 12, 41, 13, 29, 25, 54 | hdmapval2 41834 | . . 3 ⊢ (𝜑 → (𝑆‘𝑋) = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑋〉)) |
| 56 | 52 | simprd 495 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑌})) |
| 57 | 3, 6, 5, 44, 45, 30, 33, 23, 56 | hdmaplem4 41776 | . . . 4 ⊢ (𝜑 → ¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑌}))) |
| 58 | 1, 18, 2, 3, 6, 7, 8, 15, 12, 41, 13, 30, 25, 57 | hdmapval2 41834 | . . 3 ⊢ (𝜑 → (𝑆‘𝑌) = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑌〉)) |
| 59 | 55, 58 | oveq12d 7449 | . 2 ⊢ (𝜑 → ((𝑆‘𝑋) ✚ (𝑆‘𝑌)) = ((𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑋〉) ✚ (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑌〉))) |
| 60 | 40, 51, 59 | 3eqtr4d 2787 | 1 ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) ✚ (𝑆‘𝑌))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 {csn 4626 {cpr 4628 〈cop 4632 〈cotp 4634 I cid 5577 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 0gc0g 17484 -gcsg 18953 LModclmod 20858 LSubSpclss 20929 LSpanclspn 20969 HLchlt 39351 LHypclh 39986 LTrncltrn 40103 DVecHcdvh 41080 LCDualclcd 41588 mapdcmpd 41626 HVMapchvm 41758 HDMap1chdma1 41793 HDMapchdma 41794 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-riotaBAD 38954 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-undef 8298 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-0g 17486 df-mre 17629 df-mrc 17630 df-acs 17632 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-clat 18544 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cntz 19335 df-oppg 19364 df-lsm 19654 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-nzr 20513 df-rlreg 20694 df-domn 20695 df-drng 20731 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lvec 21102 df-lsatoms 38977 df-lshyp 38978 df-lcv 39020 df-lfl 39059 df-lkr 39087 df-ldual 39125 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-llines 39500 df-lplanes 39501 df-lvols 39502 df-lines 39503 df-psubsp 39505 df-pmap 39506 df-padd 39798 df-lhyp 39990 df-laut 39991 df-ldil 40106 df-ltrn 40107 df-trl 40161 df-tgrp 40745 df-tendo 40757 df-edring 40759 df-dveca 41005 df-disoa 41031 df-dvech 41081 df-dib 41141 df-dic 41175 df-dih 41231 df-doch 41350 df-djh 41397 df-lcdual 41589 df-mapd 41627 df-hvmap 41759 df-hdmap1 41795 df-hdmap 41796 |
| This theorem is referenced by: hdmap11lem2 41844 |
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