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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 30443 | . . . 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 30523 | . . 3 β’ ((π β NrmCVec β§ π΄ β π β§ π β π) β (ββ(π΄ππ)) = (πππ΄)) |
| 7 | 4, 6 | mpd3an3 1459 | . 2 β’ ((π β NrmCVec β§ π΄ β π) β (ββ(π΄ππ)) = (πππ΄)) |
| 8 | 1, 2, 5 | dip0r 30526 | . . . 4 β’ ((π β NrmCVec β§ π΄ β π) β (π΄ππ) = 0) |
| 9 | 8 | fveq2d 6901 | . . 3 β’ ((π β NrmCVec β§ π΄ β π) β (ββ(π΄ππ)) = (ββ0)) |
| 10 | cj0 15137 | . . 3 β’ (ββ0) = 0 | |
| 11 | 9, 10 | eqtrdi 2784 | . 2 β’ ((π β NrmCVec β§ π΄ β π) β (ββ(π΄ππ)) = 0) |
| 12 | 7, 11 | eqtr3d 2770 | 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 1534 β wcel 2099 βcfv 6548 (class class class)co 7420 0cc0 11138 βccj 15075 NrmCVeccnv 30393 BaseSetcba 30395 0veccn0v 30397 Β·πOLDcdip 30509 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-sup 9465 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-fz 13517 df-fzo 13660 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 df-sum 15665 df-grpo 30302 df-gid 30303 df-ginv 30304 df-ablo 30354 df-vc 30368 df-nv 30401 df-va 30404 df-ba 30405 df-sm 30406 df-0v 30407 df-nmcv 30409 df-dip 30510 |
| This theorem is referenced by: ip2i 30637 ipasslem1 30640 ipasslem2 30641 |
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