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| Mirrors > Home > MPE Home > Th. List > ip0r | Structured version Visualization version GIF version | ||
| Description: Inner product with a zero second argument. (Contributed by NM, 5-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.) |
| Ref | Expression |
|---|---|
| phlsrng.f | β’ πΉ = (Scalarβπ) |
| phllmhm.h | β’ , = (Β·πβπ) |
| phllmhm.v | β’ π = (Baseβπ) |
| ip0l.z | β’ π = (0gβπΉ) |
| ip0l.o | β’ 0 = (0gβπ) |
| Ref | Expression |
|---|---|
| ip0r | β’ ((π β PreHil β§ π΄ β π) β (π΄ , 0 ) = π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | phlsrng.f | . . . 4 β’ πΉ = (Scalarβπ) | |
| 2 | phllmhm.h | . . . 4 β’ , = (Β·πβπ) | |
| 3 | phllmhm.v | . . . 4 β’ π = (Baseβπ) | |
| 4 | ip0l.z | . . . 4 β’ π = (0gβπΉ) | |
| 5 | ip0l.o | . . . 4 β’ 0 = (0gβπ) | |
| 6 | 1, 2, 3, 4, 5 | ip0l 21499 | . . 3 β’ ((π β PreHil β§ π΄ β π) β ( 0 , π΄) = π) |
| 7 | 6 | fveq2d 6886 | . 2 β’ ((π β PreHil β§ π΄ β π) β ((*πβπΉ)β( 0 , π΄)) = ((*πβπΉ)βπ)) |
| 8 | phllmod 21493 | . . . . 5 β’ (π β PreHil β π β LMod) | |
| 9 | 8 | adantr 480 | . . . 4 β’ ((π β PreHil β§ π΄ β π) β π β LMod) |
| 10 | 3, 5 | lmod0vcl 20729 | . . . 4 β’ (π β LMod β 0 β π) |
| 11 | 9, 10 | syl 17 | . . 3 β’ ((π β PreHil β§ π΄ β π) β 0 β π) |
| 12 | eqid 2724 | . . . . . 6 β’ (*πβπΉ) = (*πβπΉ) | |
| 13 | 1, 2, 3, 12 | ipcj 21497 | . . . . 5 β’ ((π β PreHil β§ 0 β π β§ π΄ β π) β ((*πβπΉ)β( 0 , π΄)) = (π΄ , 0 )) |
| 14 | 13 | 3expa 1115 | . . . 4 β’ (((π β PreHil β§ 0 β π) β§ π΄ β π) β ((*πβπΉ)β( 0 , π΄)) = (π΄ , 0 )) |
| 15 | 14 | an32s 649 | . . 3 β’ (((π β PreHil β§ π΄ β π) β§ 0 β π) β ((*πβπΉ)β( 0 , π΄)) = (π΄ , 0 )) |
| 16 | 11, 15 | mpdan 684 | . 2 β’ ((π β PreHil β§ π΄ β π) β ((*πβπΉ)β( 0 , π΄)) = (π΄ , 0 )) |
| 17 | 1 | phlsrng 21494 | . . . 4 β’ (π β PreHil β πΉ β *-Ring) |
| 18 | 17 | adantr 480 | . . 3 β’ ((π β PreHil β§ π΄ β π) β πΉ β *-Ring) |
| 19 | 12, 4 | srng0 20695 | . . 3 β’ (πΉ β *-Ring β ((*πβπΉ)βπ) = π) |
| 20 | 18, 19 | syl 17 | . 2 β’ ((π β PreHil β§ π΄ β π) β ((*πβπΉ)βπ) = π) |
| 21 | 7, 16, 20 | 3eqtr3d 2772 | 1 β’ ((π β PreHil β§ π΄ β π) β (π΄ , 0 ) = π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 βcfv 6534 (class class class)co 7402 Basecbs 17145 *πcstv 17200 Scalarcsca 17201 Β·πcip 17203 0gc0g 17386 *-Ringcsr 20679 LModclmod 20698 PreHilcphl 21487 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-plusg 17211 df-mulr 17212 df-sca 17214 df-vsca 17215 df-ip 17216 df-0g 17388 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-grp 18858 df-ghm 19131 df-mgp 20032 df-ur 20079 df-ring 20132 df-oppr 20228 df-rhm 20366 df-staf 20680 df-srng 20681 df-lmod 20700 df-lmhm 20862 df-lvec 20943 df-sra 21013 df-rgmod 21014 df-phl 21489 |
| This theorem is referenced by: cphip0r 25055 ipcau2 25086 |
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