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| Mirrors > Home > MPE Home > Th. List > dip0l | Structured version Visualization version GIF version | ||
| Description: Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. (Contributed by NM, 5-Feb-2007.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| dip0r.1 | β’ π = (BaseSetβπ) |
| dip0r.5 | β’ π = (0vecβπ) |
| dip0r.7 | β’ π = (Β·πOLDβπ) |
| Ref | Expression |
|---|---|
| dip0l | β’ ((π β NrmCVec β§ π΄ β π) β (πππ΄) = 0) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dip0r.1 | . . . . 5 β’ π = (BaseSetβπ) | |
| 2 | dip0r.5 | . . . . 5 β’ π = (0vecβπ) | |
| 3 | 1, 2 | nvzcl 30381 | . . . 4 β’ (π β NrmCVec β π β π) |
| 4 | 3 | adantr 480 | . . 3 β’ ((π β NrmCVec β§ π΄ β π) β π β π) |
| 5 | dip0r.7 | . . . 4 β’ π = (Β·πOLDβπ) | |
| 6 | 1, 5 | dipcj 30461 | . . 3 β’ ((π β NrmCVec β§ π΄ β π β§ π β π) β (ββ(π΄ππ)) = (πππ΄)) |
| 7 | 4, 6 | mpd3an3 1458 | . 2 β’ ((π β NrmCVec β§ π΄ β π) β (ββ(π΄ππ)) = (πππ΄)) |
| 8 | 1, 2, 5 | dip0r 30464 | . . . 4 β’ ((π β NrmCVec β§ π΄ β π) β (π΄ππ) = 0) |
| 9 | 8 | fveq2d 6886 | . . 3 β’ ((π β NrmCVec β§ π΄ β π) β (ββ(π΄ππ)) = (ββ0)) |
| 10 | cj0 15107 | . . 3 β’ (ββ0) = 0 | |
| 11 | 9, 10 | eqtrdi 2780 | . 2 β’ ((π β NrmCVec β§ π΄ β π) β (ββ(π΄ππ)) = 0) |
| 12 | 7, 11 | eqtr3d 2766 | 1 β’ ((π β NrmCVec β§ π΄ β π) β (πππ΄) = 0) |
| 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 0cc0 11107 βccj 15045 NrmCVeccnv 30331 BaseSetcba 30333 0veccn0v 30335 Β·πOLDcdip 30447 |
| 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-inf2 9633 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 ax-pre-sup 11185 |
| 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-int 4942 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-se 5623 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-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-oi 9502 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-n0 12472 df-z 12558 df-uz 12822 df-rp 12976 df-fz 13486 df-fzo 13629 df-seq 13968 df-exp 14029 df-hash 14292 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-sum 15635 df-grpo 30240 df-gid 30241 df-ginv 30242 df-ablo 30292 df-vc 30306 df-nv 30339 df-va 30342 df-ba 30343 df-sm 30344 df-0v 30345 df-nmcv 30347 df-dip 30448 |
| This theorem is referenced by: ip2i 30575 ipasslem1 30578 ipasslem2 30579 |
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