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Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dvhlveclem | Structured version Visualization version Unicode version | ||
Description: Lemma for dvhlvec 36398. TODO: proof substituting inner part first
shorter/longer than substituting outer part first? TODO: break up into
smaller lemmas? TODO: does   method shorten proof?
(Contributed by NM, 22-Oct-2013.) (Proof shortened by Mario Carneiro,
24-Jun-2014.) |
| Ref | Expression |
|---|---|
| dvhgrp.b |
        |
| dvhgrp.h |
  |
| dvhgrp.t |
   |
| dvhgrp.e |
  |
| dvhgrp.u |
  |
| dvhgrp.d |
 Scalar |
| dvhgrp.p |
  |
| dvhgrp.a |
 |
| dvhgrp.o |
  |
| dvhgrp.i |
   |
| dvhlvec.m |
  |
| dvhlvec.s |
  |
| Ref | Expression |
|---|---|
| dvhlveclem |
  ![/\](wedge.gif "/\"){kind=small}
 |
are
mutually distinct and distinct from all other variables.| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dvhgrp.h |
. . . . 5
| |
| 2 | dvhgrp.t |
. . . . 5
| |
| 3 | dvhgrp.e |
. . . . 5
| |
| 4 | dvhgrp.u |
. . . . 5
| |
| 5 | eqid 2622 |
. . . . 5
| |
| 6 | 1, 2, 3, 4, 5 | dvhvbase 36376 |
. . . 4
 |
| 7 | 6 | eqcomd 2628 |
. . 3
|
| 8 | dvhgrp.a |
. . . 4
| |
| 9 | 8 | a1i 11 |
. . 3
|
| 10 | dvhgrp.d |
. . . 4
Scalar | |
| 11 | 10 | a1i 11 |
. . 3
Scalar |
| 12 | dvhlvec.s |
. . . 4
| |
| 13 | 12 | a1i 11 |
. . 3
|
| 14 | eqid 2622 |
. . . . 5
| |
| 15 | 1, 3, 4, 10, 14 | dvhbase 36372 |
. . . 4
|
| 16 | 15 | eqcomd 2628 |
. . 3
|
| 17 | dvhgrp.p |
. . . 4
| |
| 18 | 17 | a1i 11 |
. . 3
|
| 19 | dvhlvec.m |
. . . 4
| |
| 20 | 19 | a1i 11 |
. . 3
|
| 21 | eqid 2622 |
. . . . . 6
 | |
| 22 | 1, 21, 4, 10 | dvhsca 36371 |
. . . . 5
|
| 23 | 22 | fveq2d 6195 |
. . . 4
 |
| 24 | eqid 2622 |
. . . . 5
| |
| 25 | 1, 2, 21, 24 | erng1r 36283 |
. . . 4
  |
| 26 | 23, 25 | eqtr2d 2657 |
. . 3
|
| 27 | 1, 21 | erngdv 36281 |
. . . . 5
 |
| 28 | 22, 27 | eqeltrd 2701 |
. . . 4
|
| 29 | drngring 18754 |
. . . 4
 | |
| 30 | 28, 29 | syl 17 |
. . 3
|
| 31 | dvhgrp.b |
. . . 4
| |
| 32 | dvhgrp.o |
. . . 4
| |
| 33 | dvhgrp.i |
. . . 4
| |
| 34 | 31, 1, 2, 3, 4, 10, 17, 8, 32, 33 | dvhgrp 36396 |
. . 3
 |
| 35 | 1, 2, 3, 4, 12 | dvhvscacl 36392 |
. . . 4
|
| 36 | 35 | 3impb 1260 |
. . 3
|
| 37 | simpl 473 |
. . . . . . 7
| |
| 38 | simpr1 1067 |
. . . . . . 7
| |
| 39 | simpr2 1068 |
. . . . . . . 8
| |
| 40 | xp1st 7198 |
. . . . . . . 8
 | |
| 41 | 39, 40 | syl 17 |
. . . . . . 7
|
| 42 | simpr3 1069 |
. . . . . . . 8
| |
| 43 | xp1st 7198 |
. . . . . . . 8
| |
| 44 | 42, 43 | syl 17 |
. . . . . . 7
|
| 45 | 1, 2, 3 | tendospdi1 36309 |
. . . . . . 7
 |
| 46 | 37, 38, 41, 44, 45 | syl13anc 1328 |
. . . . . 6
|
| 47 | 1, 2, 3, 4, 10, 8, 17 | dvhvadd 36381 |
. . . . . . . . . 10
    |
| 48 | 47 | 3adantr1 1220 |
. . . . . . . . 9
|
| 49 | 48 | fveq2d 6195 |
. . . . . . . 8
|
| 50 | fvex 6201 |
. . . . . . . . . 10
 | |
| 51 | fvex 6201 |
. . . . . . . . . 10
| |
| 52 | 50, 51 | coex 7118 |
. . . . . . . . 9
|
| 53 | ovex 6678 |
. . . . . . . . 9
| |
| 54 | 52, 53 | op1st 7176 |
. . . . . . . 8
|
| 55 | 49, 54 | syl6eq 2672 |
. . . . . . 7
|
| 56 | 55 | fveq2d 6195 |
. . . . . 6
|
| 57 | 1, 2, 3, 4, 12 | dvhvsca 36390 |
. . . . . . . . . 10
|
| 58 | 57 | 3adantr3 1222 |
. . . . . . . . 9
|
| 59 | 58 | fveq2d 6195 |
. . . . . . . 8
|
| 60 | fvex 6201 |
. . . . . . . . 9
| |
| 61 | vex 3203 |
. . . . . . . . . 10
| |
| 62 | fvex 6201 |
. . . . . . . . . 10
| |
| 63 | 61, 62 | coex 7118 |
. . . . . . . . 9
|
| 64 | 60, 63 | op1st 7176 |
. . . . . . . 8
|
| 65 | 59, 64 | syl6eq 2672 |
. . . . . . 7
|
| 66 | 1, 2, 3, 4, 12 | dvhvsca 36390 |
. . . . . . . . . 10
|
| 67 | 66 | 3adantr2 1221 |
. . . . . . . . 9
|
| 68 | 67 | fveq2d 6195 |
. . . . . . . 8
|
| 69 | fvex 6201 |
. . . . . . . . 9
| |
| 70 | fvex 6201 |
. . . . . . . . . 10
| |
| 71 | 61, 70 | coex 7118 |
. . . . . . . . 9
|
| 72 | 69, 71 | op1st 7176 |
. . . . . . . 8
|
| 73 | 68, 72 | syl6eq 2672 |
. . . . . . 7
|
| 74 | 65, 73 | coeq12d 5286 |
. . . . . 6
|
| 75 | 46, 56, 74 | 3eqtr4d 2666 |
. . . . 5
|
| 76 | 30 | adantr 481 |
. . . . . . . 8
|
| 77 | 16 | adantr 481 |
. . . . . . . . 9
|
| 78 | 38, 77 | eleqtrd 2703 |
. . . . . . . 8
|
| 79 | xp2nd 7199 |
. . . . . . . . . 10
| |
| 80 | 39, 79 | syl 17 |
. . . . . . . . 9
|
| 81 | 80, 77 | eleqtrd 2703 |
. . . . . . . 8
|
| 82 | xp2nd 7199 |
. . . . . . . . . 10
| |
| 83 | 42, 82 | syl 17 |
. . . . . . . . 9
|
| 84 | 83, 77 | eleqtrd 2703 |
. . . . . . . 8
|
| 85 | 14, 17, 19 | ringdi 18566 |
. . . . . . . 8
|
| 86 | 76, 78, 81, 84, 85 | syl13anc 1328 |
. . . . . . 7
|
| 87 | 14, 17 | ringacl 18578 |
. . . . . . . . . 10
|
| 88 | 76, 81, 84, 87 | syl3anc 1326 |
. . . . . . . . 9
|
| 89 | 88, 77 | eleqtrrd 2704 |
. . . . . . . 8
|
| 90 | 1, 2, 3, 4, 10, 19 | dvhmulr 36375 |
. . . . . . . 8
|
| 91 | 37, 38, 89, 90 | syl12anc 1324 |
. . . . . . 7
|
| 92 | 1, 2, 3, 4, 10, 19 | dvhmulr 36375 |
. . . . . . . . 9
|
| 93 | 37, 38, 80, 92 | syl12anc 1324 |
. . . . . . . 8
|
| 94 | 1, 2, 3, 4, 10, 19 | dvhmulr 36375 |
. . . . . . . . 9
|
| 95 | 37, 38, 83, 94 | syl12anc 1324 |
. . . . . . . 8
|
| 96 | 93, 95 | oveq12d 6668 |
. . . . . . 7
|
| 97 | 86, 91, 96 | 3eqtr3d 2664 |
. . . . . 6
|
| 98 | 48 | fveq2d 6195 |
. . . . . . . 8
|
| 99 | 52, 53 | op2nd 7177 |
. . . . . . . 8
|
| 100 | 98, 99 | syl6eq 2672 |
. . . . . . 7
|
| 101 | 100 | coeq2d 5284 |
. . . . . 6
|
| 102 | 58 | fveq2d 6195 |
. . . . . . . 8
|
| 103 | 60, 63 | op2nd 7177 |
. . . . . . . 8
|
| 104 | 102, 103 | syl6eq 2672 |
. . . . . . 7
|
| 105 | 67 | fveq2d 6195 |
. . . . . . . 8
|
| 106 | 69, 71 | op2nd 7177 |
. . . . . . . 8
|
| 107 | 105, 106 | syl6eq 2672 |
. . . . . . 7
|
| 108 | 104, 107 | oveq12d 6668 |
. . . . . 6
|
| 109 | 97, 101, 108 | 3eqtr4d 2666 |
. . . . 5
|
| 110 | 75, 109 | opeq12d 4410 |
. . . 4
|
| 111 | 1, 2, 3, 4, 10, 17, 8 | dvhvaddcl 36384 |
. . . . . 6
|
| 112 | 111 | 3adantr1 1220 |
. . . . 5
|
| 113 | 1, 2, 3, 4, 12 | dvhvsca 36390 |
. . . . 5
|
| 114 | 37, 38, 112, 113 | syl12anc 1324 |
. . . 4
|
| 115 | 35 | 3adantr3 1222 |
. . . . 5
|
| 116 | 1, 2, 3, 4, 12 | dvhvscacl 36392 |
. . . . . 6
|
| 117 | 116 | 3adantr2 1221 |
. . . . 5
|
| 118 | 1, 2, 3, 4, 10, 8, 17 | dvhvadd 36381 |
. . . . 5
|
| 119 | 37, 115, 117, 118 | syl12anc 1324 |
. . . 4
|
| 120 | 110, 114, 119 | 3eqtr4d 2666 |
. . 3
|
| 121 | simpl 473 |
. . . . . . 7
| |
| 122 | simpr1 1067 |
. . . . . . 7
| |
| 123 | simpr2 1068 |
. . . . . . 7
| |
| 124 | simpr3 1069 |
. . . . . . . 8
| |
| 125 | 124, 43 | syl 17 |
. . . . . . 7
|
| 126 | eqid 2622 |
. . . . . . . 8
| |
| 127 | 1, 2, 3, 21, 126 | erngplus2 36092 |
. . . . . . 7
|
| 128 | 121, 122, 123, 125, 127 | syl13anc 1328 |
. . . . . 6
|
| 129 | 22 | fveq2d 6195 |
. . . . . . . . . 10
|
| 130 | 17, 129 | syl5eq 2668 |
. . . . . . . . 9
|
| 131 | 130 | oveqd 6667 |
. . . . . . . 8
|
| 132 | 131 | fveq1d 6193 |
. . . . . . 7
|
| 133 | 132 | adantr 481 |
. . . . . 6
|
| 134 | 66 | 3adantr2 1221 |
. . . . . . . . 9
|
| 135 | 134 | fveq2d 6195 |
. . . . . . . 8
|
| 136 | 135, 72 | syl6eq 2672 |
. . . . . . 7
|
| 137 | 1, 2, 3, 4, 12 | dvhvsca 36390 |
. . . . . . . . . 10
|
| 138 | 137 | 3adantr1 1220 |
. . . . . . . . 9
|
| 139 | 138 | fveq2d 6195 |
. . . . . . . 8
|
| 140 | fvex 6201 |
. . . . . . . . 9
| |
| 141 | vex 3203 |
. . . . . . . . . 10
| |
| 142 | 141, 70 | coex 7118 |
. . . . . . . . 9
|
| 143 | 140, 142 | op1st 7176 |
. . . . . . . 8
|
| 144 | 139, 143 | syl6eq 2672 |
. . . . . . 7
|
| 145 | 136, 144 | coeq12d 5286 |
. . . . . 6
|
| 146 | 128, 133, 145 | 3eqtr4d 2666 |
. . . . 5
|
| 147 | 30 | adantr 481 |
. . . . . . . 8
|
| 148 | 16 | adantr 481 |
. . . . . . . . 9
|
| 149 | 122, 148 | eleqtrd 2703 |
. . . . . . . 8
|
| 150 | 123, 148 | eleqtrd 2703 |
. . . . . . . 8
|
| 151 | 124, 82 | syl 17 |
. . . . . . . . 9
|
| 152 | 151, 148 | eleqtrd 2703 |
. . . . . . . 8
|
| 153 | 14, 17, 19 | ringdir 18567 |
. . . . . . . 8
|
| 154 | 147, 149, 150, 152, 153 | syl13anc 1328 |
. . . . . . 7
|
| 155 | 14, 17 | ringacl 18578 |
. . . . . . . . . 10
|
| 156 | 147, 149, 150, 155 | syl3anc 1326 |
. . . . . . . . 9
|
| 157 | 156, 148 | eleqtrrd 2704 |
. . . . . . . 8
|
| 158 | 1, 2, 3, 4, 10, 19 | dvhmulr 36375 |
. . . . . . . 8
|
| 159 | 121, 157, 151, 158 | syl12anc 1324 |
. . . . . . 7
|
| 160 | 121, 122, 151, 94 | syl12anc 1324 |
. . . . . . . 8
|
| 161 | 1, 2, 3, 4, 10, 19 | dvhmulr 36375 |
. . . . . . . . 9
|
| 162 | 121, 123, 151, 161 | syl12anc 1324 |
. . . . . . . 8
|
| 163 | 160, 162 | oveq12d 6668 |
. . . . . . 7
|
| 164 | 154, 159, 163 | 3eqtr3d 2664 |
. . . . . 6
|
| 165 | 134 | fveq2d 6195 |
. . . . . . . 8
|
| 166 | 165, 106 | syl6eq 2672 |
. . . . . . 7
|
| 167 | 138 | fveq2d 6195 |
. . . . . . . 8
|
| 168 | 140, 142 | op2nd 7177 |
. . . . . . . 8
|
| 169 | 167, 168 | syl6eq 2672 |
. . . . . . 7
|
| 170 | 166, 169 | oveq12d 6668 |
. . . . . 6
|
| 171 | 164, 170 | eqtr4d 2659 |
. . . . 5
|
| 172 | 146, 171 | opeq12d 4410 |
. . . 4
|
| 173 | 1, 2, 3, 4, 12 | dvhvsca 36390 |
. . . . 5
|
| 174 | 121, 157, 124, 173 | syl12anc 1324 |
. . . 4
|
| 175 | 116 | 3adantr2 1221 |
. . . . 5
|
| 176 | 1, 2, 3, 4, 12 | dvhvscacl 36392 |
. . . . . 6
|
| 177 | 176 | 3adantr1 1220 |
. . . . 5
|
| 178 | 1, 2, 3, 4, 10, 8, 17 | dvhvadd 36381 |
. . . . 5
|
| 179 | 121, 175, 177, 178 | syl12anc 1324 |
. . . 4
|
| 180 | 172, 174, 179 | 3eqtr4d 2666 |
. . 3
|
| 181 | 1, 2, 3 | tendocoval 36054 |
. . . . . . 7
|
| 182 | 121, 122, 123, 125, 181 | syl121anc 1331 |
. . . . . 6
|
| 183 | coass 5654 |
. . . . . . 7
| |
| 184 | 183 | a1i 11 |
. . . . . 6
|
| 185 | 182, 184 | opeq12d 4410 |
. . . . 5
|
| 186 | 1, 3 | tendococl 36060 |
. . . . . . 7
|
| 187 | 121, 122, 123, 186 | syl3anc 1326 |
. . . . . 6
|
| 188 | 1, 2, 3, 4, 12 | dvhvsca 36390 |
. . . . . 6
|
| 189 | 121, 187, 124, 188 | syl12anc 1324 |
. . . . 5
|
| 190 | 1, 2, 3 | tendocl 36055 |
. . . . . . 7
|
| 191 | 121, 123, 125, 190 | syl3anc 1326 |
. . . . . 6
|
| 192 | 1, 3 | tendococl 36060 |
. . . . . . 7
|
| 193 | 121, 123, 151, 192 | syl3anc 1326 |
. . . . . 6
|
| 194 | 1, 2, 3, 4, 12 | dvhopvsca 36391 |
. . . . . 6
|
| 195 | 121, 122, 191, 193, 194 | syl13anc 1328 |
. . . . 5
|
| 196 | 185, 189, 195 | 3eqtr4d 2666 |
. . . 4
|
| 197 | 1, 2, 3, 4, 10, 19 | dvhmulr 36375 |
. . . . . 6
|
| 198 | 197 | 3adantr3 1222 |
. . . . 5
|
| 199 | 198 | oveq1d 6665 |
. . . 4
|
| 200 | 138 | oveq2d 6666 |
. . . 4
|
| 201 | 196, 199, 200 | 3eqtr4d 2666 |
. . 3
|
| 202 | xp1st 7198 |
. . . . . . 7
| |
| 203 | 202 | adantl 482 |
. . . . . 6
|
| 204 | tendospid 36306 |
. . . . . 6
| |
| 205 | 203, 204 | syl 17 |
. . . . 5
|
| 206 | xp2nd 7199 |
. . . . . . 7
| |
| 207 | 1, 2, 3 | tendof 36051 |
. . . . . . 7
  |
| 208 | 206, 207 | sylan2 491 |
. . . . . 6
|
| 209 | fcoi2 6079 |
. . . . . 6
| |
| 210 | 208, 209 | syl 17 |
. . . . 5
|
| 211 | 205, 210 | opeq12d 4410 |
. . . 4
|
| 212 | 1, 2, 3 | tendoidcl 36057 |
. . . . . 6
|
| 213 | 212 | anim1i 592 |
. . . . 5
|
| 214 | 1, 2, 3, 4, 12 | dvhvsca 36390 |
. . . . 5
|
| 215 | 213, 214 | syldan 487 |
. . . 4
|
| 216 | 1st2nd2 7205 |
. . . . 5
| |
| 217 | 216 | adantl 482 |
. . . 4
|
| 218 | 211, 215, 217 | 3eqtr4d 2666 |
. . 3
|
| 219 | 7, 9, 11, 13, 16, 18, 20, 26, 30, 34, 36, 120, 180, 201, 218 | islmodd 18869 |
. 2
 |
| 220 | 10 | islvec 19104 |
. 2
 |
| 221 | 219, 28, 220 | sylanbrc 698 |
1
|
| Colors of variables: wff setvar class |
| Syntax hints: wi 4
wa 384
w3a 1037
wceq 1483
wcel 1990
cop 4183
cid 5023
cxp 5112
cres 5116
ccom 5118
wf 5884 cfv 5888 (class class class)co 6650
c1st 7166
c2nd 7167
cbs 15857
cplusg 15941 cmulr 15942 Scalarcsca 15944
cvsca 15945 c0g 16100 cminusg 17423 cur 18501 crg 18547 cdr 18747
clmod 18863 clvec 19102 chlt 34637 clh 35270 cltrn 35387 ctendo 36040 cedring 36041 cdvh 36367 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-riotaBAD 34239 |
| This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-undef 7399 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-0g 16102 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-lmod 18865 df-lvec 19103 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lvols 34786 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446 df-tendo 36043 df-edring 36045 df-dvech 36368 |
| This theorem is referenced by: dvhlvec 36398 |
| Copyright terms: Public domain | W3C validator |