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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap14lem10 | Structured version Visualization version GIF version |
Description: Part of proof of part 14 in [Baer] p. 49 line 38. (Contributed by NM, 3-Jun-2015.) |
Ref | Expression |
---|---|
hdmap14lem8.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap14lem8.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap14lem8.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap14lem8.q | ⊢ + = (+g‘𝑈) |
hdmap14lem8.t | ⊢ · = ( ·𝑠 ‘𝑈) |
hdmap14lem8.o | ⊢ 0 = (0g‘𝑈) |
hdmap14lem8.n | ⊢ 𝑁 = (LSpan‘𝑈) |
hdmap14lem8.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hdmap14lem8.b | ⊢ 𝐵 = (Base‘𝑅) |
hdmap14lem8.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmap14lem8.d | ⊢ ✚ = (+g‘𝐶) |
hdmap14lem8.e | ⊢ ∙ = ( ·𝑠 ‘𝐶) |
hdmap14lem8.p | ⊢ 𝑃 = (Scalar‘𝐶) |
hdmap14lem8.a | ⊢ 𝐴 = (Base‘𝑃) |
hdmap14lem8.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmap14lem8.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap14lem8.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
hdmap14lem8.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
hdmap14lem8.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
hdmap14lem8.g | ⊢ (𝜑 → 𝐺 ∈ 𝐴) |
hdmap14lem8.i | ⊢ (𝜑 → 𝐼 ∈ 𝐴) |
hdmap14lem8.xx | ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) |
hdmap14lem8.yy | ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑌)) = (𝐼 ∙ (𝑆‘𝑌))) |
hdmap14lem8.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
Ref | Expression |
---|---|
hdmap14lem10 | ⊢ (𝜑 → 𝐺 = 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap14lem8.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmap14lem8.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmap14lem8.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
4 | hdmap14lem8.t | . . 3 ⊢ · = ( ·𝑠 ‘𝑈) | |
5 | hdmap14lem8.r | . . 3 ⊢ 𝑅 = (Scalar‘𝑈) | |
6 | hdmap14lem8.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
7 | hdmap14lem8.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
8 | hdmap14lem8.e | . . 3 ⊢ ∙ = ( ·𝑠 ‘𝐶) | |
9 | eqid 2732 | . . 3 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶) | |
10 | hdmap14lem8.p | . . 3 ⊢ 𝑃 = (Scalar‘𝐶) | |
11 | hdmap14lem8.a | . . 3 ⊢ 𝐴 = (Base‘𝑃) | |
12 | hdmap14lem8.s | . . 3 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
13 | hdmap14lem8.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
14 | 1, 2, 13 | dvhlmod 39969 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
15 | hdmap14lem8.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
16 | 15 | eldifad 3959 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
17 | hdmap14lem8.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
18 | 17 | eldifad 3959 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
19 | hdmap14lem8.q | . . . . 5 ⊢ + = (+g‘𝑈) | |
20 | 3, 19 | lmodvacl 20478 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉) |
21 | 14, 16, 18, 20 | syl3anc 1371 | . . 3 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉) |
22 | hdmap14lem8.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
23 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 21, 22 | hdmap14lem2a 40726 | . 2 ⊢ (𝜑 → ∃𝑔 ∈ 𝐴 (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) |
24 | hdmap14lem8.o | . . . 4 ⊢ 0 = (0g‘𝑈) | |
25 | hdmap14lem8.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑈) | |
26 | hdmap14lem8.d | . . . 4 ⊢ ✚ = (+g‘𝐶) | |
27 | 13 | 3ad2ant1 1133 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
28 | 15 | 3ad2ant1 1133 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
29 | 17 | 3ad2ant1 1133 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
30 | 22 | 3ad2ant1 1133 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝐹 ∈ 𝐵) |
31 | hdmap14lem8.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐴) | |
32 | 31 | 3ad2ant1 1133 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝐺 ∈ 𝐴) |
33 | hdmap14lem8.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝐴) | |
34 | 33 | 3ad2ant1 1133 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝐼 ∈ 𝐴) |
35 | hdmap14lem8.xx | . . . . 5 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) | |
36 | 35 | 3ad2ant1 1133 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) |
37 | hdmap14lem8.yy | . . . . 5 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑌)) = (𝐼 ∙ (𝑆‘𝑌))) | |
38 | 37 | 3ad2ant1 1133 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → (𝑆‘(𝐹 · 𝑌)) = (𝐼 ∙ (𝑆‘𝑌))) |
39 | hdmap14lem8.ne | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
40 | 39 | 3ad2ant1 1133 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
41 | simp2 1137 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝑔 ∈ 𝐴) | |
42 | simp3 1138 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) | |
43 | 1, 2, 3, 19, 4, 24, 25, 5, 6, 7, 26, 8, 10, 11, 12, 27, 28, 29, 30, 32, 34, 36, 38, 40, 41, 42 | hdmap14lem9 40735 | . . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐴 ∧ (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌)))) → 𝐺 = 𝐼) |
44 | 43 | rexlimdv3a 3159 | . 2 ⊢ (𝜑 → (∃𝑔 ∈ 𝐴 (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑔 ∙ (𝑆‘(𝑋 + 𝑌))) → 𝐺 = 𝐼)) |
45 | 23, 44 | mpd 15 | 1 ⊢ (𝜑 → 𝐺 = 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∃wrex 3070 ∖ cdif 3944 {csn 4627 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 +gcplusg 17193 Scalarcsca 17196 ·𝑠 cvsca 17197 0gc0g 17381 LModclmod 20463 LSpanclspn 20574 HLchlt 38208 LHypclh 38843 DVecHcdvh 39937 LCDualclcd 40445 HDMapchdma 40651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-riotaBAD 37811 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-ot 4636 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8207 df-undef 8254 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-0g 17383 df-mre 17526 df-mrc 17527 df-acs 17529 df-proset 18244 df-poset 18262 df-plt 18279 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-p1 18375 df-lat 18381 df-clat 18448 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-cntz 19175 df-oppg 19204 df-lsm 19498 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-invr 20194 df-dvr 20207 df-drng 20309 df-lmod 20465 df-lss 20535 df-lsp 20575 df-lvec 20706 df-lsatoms 37834 df-lshyp 37835 df-lcv 37877 df-lfl 37916 df-lkr 37944 df-ldual 37982 df-oposet 38034 df-ol 38036 df-oml 38037 df-covers 38124 df-ats 38125 df-atl 38156 df-cvlat 38180 df-hlat 38209 df-llines 38357 df-lplanes 38358 df-lvols 38359 df-lines 38360 df-psubsp 38362 df-pmap 38363 df-padd 38655 df-lhyp 38847 df-laut 38848 df-ldil 38963 df-ltrn 38964 df-trl 39018 df-tgrp 39602 df-tendo 39614 df-edring 39616 df-dveca 39862 df-disoa 39888 df-dvech 39938 df-dib 39998 df-dic 40032 df-dih 40088 df-doch 40207 df-djh 40254 df-lcdual 40446 df-mapd 40484 df-hvmap 40616 df-hdmap1 40652 df-hdmap 40653 |
This theorem is referenced by: hdmap14lem11 40737 |
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