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| Mirrors > Home > MPE Home > Th. List > ipval2lem4 | Structured version Visualization version GIF version | ||
| Description: Lemma for ipval3 29950. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| dipfval.1 | β’ π = (BaseSetβπ) |
| dipfval.2 | β’ πΊ = ( +π£ βπ) |
| dipfval.4 | β’ π = ( Β·π OLD βπ) |
| dipfval.6 | β’ π = (normCVβπ) |
| dipfval.7 | β’ π = (Β·πOLDβπ) |
| Ref | Expression |
|---|---|
| ipval2lem4 | β’ (((π β NrmCVec β§ π΄ β π β§ π΅ β π) β§ πΆ β β) β ((πβ(π΄πΊ(πΆππ΅)))β2) β β) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dipfval.1 | . . 3 β’ π = (BaseSetβπ) | |
| 2 | dipfval.2 | . . 3 β’ πΊ = ( +π£ βπ) | |
| 3 | dipfval.4 | . . 3 β’ π = ( Β·π OLD βπ) | |
| 4 | dipfval.6 | . . 3 β’ π = (normCVβπ) | |
| 5 | dipfval.7 | . . 3 β’ π = (Β·πOLDβπ) | |
| 6 | 1, 2, 3, 4, 5 | ipval2lem2 29945 | . 2 β’ (((π β NrmCVec β§ π΄ β π β§ π΅ β π) β§ πΆ β β) β ((πβ(π΄πΊ(πΆππ΅)))β2) β β) |
| 7 | 6 | recnd 11239 | 1 β’ (((π β NrmCVec β§ π΄ β π β§ π΅ β π) β§ πΆ β β) β ((πβ(π΄πΊ(πΆππ΅)))β2) β β) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βcfv 6541 (class class class)co 7406 βcc 11105 2c2 12264 βcexp 14024 NrmCVeccnv 29825 +π£ cpv 29826 BaseSetcba 29827 Β·π OLD cns 29828 normCVcnmcv 29831 Β·πOLDcdip 29941 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 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 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-n0 12470 df-z 12556 df-uz 12820 df-seq 13964 df-exp 14025 df-grpo 29734 df-ablo 29786 df-vc 29800 df-nv 29833 df-va 29836 df-ba 29837 df-sm 29838 df-0v 29839 df-nmcv 29841 |
| This theorem is referenced by: ipval2 29948 ipidsq 29951 dipcl 29953 dipcj 29955 dip0r 29958 |
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