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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmapeq0 | Structured version Visualization version GIF version |
Description: Part of proof of part 12 in [Baer] p. 49 line 3. (Contributed by NM, 22-May-2015.) |
Ref | Expression |
---|---|
hdmap12a.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap12a.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap12a.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap12a.o | ⊢ 0 = (0g‘𝑈) |
hdmap12a.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmap12a.q | ⊢ 𝑄 = (0g‘𝐶) |
hdmap12a.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmap12a.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap12a.x | ⊢ (𝜑 → 𝑇 ∈ 𝑉) |
Ref | Expression |
---|---|
hdmapeq0 | ⊢ (𝜑 → ((𝑆‘𝑇) = 𝑄 ↔ 𝑇 = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap12a.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmap12a.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmap12a.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
4 | eqid 2735 | . . . . 5 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈) | |
5 | hdmap12a.c | . . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
6 | eqid 2735 | . . . . 5 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶) | |
7 | eqid 2735 | . . . . 5 ⊢ ((mapd‘𝐾)‘𝑊) = ((mapd‘𝐾)‘𝑊) | |
8 | hdmap12a.s | . . . . 5 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
9 | hdmap12a.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | hdmap12a.x | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑉) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | hdmap10 41823 | . . . 4 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘((LSpan‘𝑈)‘{𝑇})) = ((LSpan‘𝐶)‘{(𝑆‘𝑇)})) |
12 | hdmap12a.o | . . . . 5 ⊢ 0 = (0g‘𝑈) | |
13 | hdmap12a.q | . . . . 5 ⊢ 𝑄 = (0g‘𝐶) | |
14 | 1, 7, 2, 12, 5, 13, 9 | mapd0 41648 | . . . 4 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘{ 0 }) = {𝑄}) |
15 | 11, 14 | eqeq12d 2751 | . . 3 ⊢ (𝜑 → ((((mapd‘𝐾)‘𝑊)‘((LSpan‘𝑈)‘{𝑇})) = (((mapd‘𝐾)‘𝑊)‘{ 0 }) ↔ ((LSpan‘𝐶)‘{(𝑆‘𝑇)}) = {𝑄})) |
16 | eqid 2735 | . . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
17 | 1, 2, 9 | dvhlmod 41093 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
18 | 3, 16, 4 | lspsncl 20993 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑇 ∈ 𝑉) → ((LSpan‘𝑈)‘{𝑇}) ∈ (LSubSp‘𝑈)) |
19 | 17, 10, 18 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ((LSpan‘𝑈)‘{𝑇}) ∈ (LSubSp‘𝑈)) |
20 | 12, 16 | lsssn0 20964 | . . . . 5 ⊢ (𝑈 ∈ LMod → { 0 } ∈ (LSubSp‘𝑈)) |
21 | 17, 20 | syl 17 | . . . 4 ⊢ (𝜑 → { 0 } ∈ (LSubSp‘𝑈)) |
22 | 1, 2, 16, 7, 9, 19, 21 | mapd11 41622 | . . 3 ⊢ (𝜑 → ((((mapd‘𝐾)‘𝑊)‘((LSpan‘𝑈)‘{𝑇})) = (((mapd‘𝐾)‘𝑊)‘{ 0 }) ↔ ((LSpan‘𝑈)‘{𝑇}) = { 0 })) |
23 | 1, 5, 9 | lcdlmod 41575 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
24 | eqid 2735 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
25 | 1, 2, 3, 5, 24, 8, 9, 10 | hdmapcl 41813 | . . . 4 ⊢ (𝜑 → (𝑆‘𝑇) ∈ (Base‘𝐶)) |
26 | 24, 13, 6 | lspsneq0 21028 | . . . 4 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑇) ∈ (Base‘𝐶)) → (((LSpan‘𝐶)‘{(𝑆‘𝑇)}) = {𝑄} ↔ (𝑆‘𝑇) = 𝑄)) |
27 | 23, 25, 26 | syl2anc 584 | . . 3 ⊢ (𝜑 → (((LSpan‘𝐶)‘{(𝑆‘𝑇)}) = {𝑄} ↔ (𝑆‘𝑇) = 𝑄)) |
28 | 15, 22, 27 | 3bitr3rd 310 | . 2 ⊢ (𝜑 → ((𝑆‘𝑇) = 𝑄 ↔ ((LSpan‘𝑈)‘{𝑇}) = { 0 })) |
29 | 3, 12, 4 | lspsneq0 21028 | . . 3 ⊢ ((𝑈 ∈ LMod ∧ 𝑇 ∈ 𝑉) → (((LSpan‘𝑈)‘{𝑇}) = { 0 } ↔ 𝑇 = 0 )) |
30 | 17, 10, 29 | syl2anc 584 | . 2 ⊢ (𝜑 → (((LSpan‘𝑈)‘{𝑇}) = { 0 } ↔ 𝑇 = 0 )) |
31 | 28, 30 | bitrd 279 | 1 ⊢ (𝜑 → ((𝑆‘𝑇) = 𝑄 ↔ 𝑇 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {csn 4631 ‘cfv 6563 Basecbs 17245 0gc0g 17486 LModclmod 20875 LSubSpclss 20947 LSpanclspn 20987 HLchlt 39332 LHypclh 39967 DVecHcdvh 41061 LCDualclcd 41569 mapdcmpd 41607 HDMapchdma 41775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-riotaBAD 38935 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-undef 8297 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-0g 17488 df-mre 17631 df-mrc 17632 df-acs 17634 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18490 df-clat 18557 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cntz 19348 df-oppg 19377 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-nzr 20530 df-rlreg 20711 df-domn 20712 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 df-lsatoms 38958 df-lshyp 38959 df-lcv 39001 df-lfl 39040 df-lkr 39068 df-ldual 39106 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-llines 39481 df-lplanes 39482 df-lvols 39483 df-lines 39484 df-psubsp 39486 df-pmap 39487 df-padd 39779 df-lhyp 39971 df-laut 39972 df-ldil 40087 df-ltrn 40088 df-trl 40142 df-tgrp 40726 df-tendo 40738 df-edring 40740 df-dveca 40986 df-disoa 41012 df-dvech 41062 df-dib 41122 df-dic 41156 df-dih 41212 df-doch 41331 df-djh 41378 df-lcdual 41570 df-mapd 41608 df-hvmap 41740 df-hdmap1 41776 df-hdmap 41777 |
This theorem is referenced by: hdmapnzcl 41828 hdmapneg 41829 hdmap11 41831 hgmapval0 41875 hgmapval1 41876 hgmapadd 41877 hgmapmul 41878 hgmaprnlem1N 41879 hdmaplkr 41896 |
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