Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > ipval2lem3 | Structured version Visualization version GIF version | ||
| Description: Lemma for ipval3 29957. (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 |
|---|---|
| ipval2lem3 | β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β ((πβ(π΄πΊπ΅))β2) β β) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dipfval.1 | . . . . . . 7 β’ π = (BaseSetβπ) | |
| 2 | dipfval.4 | . . . . . . 7 β’ π = ( Β·π OLD βπ) | |
| 3 | 1, 2 | nvsid 29875 | . . . . . 6 β’ ((π β NrmCVec β§ π΅ β π) β (1ππ΅) = π΅) |
| 4 | 3 | oveq2d 7424 | . . . . 5 β’ ((π β NrmCVec β§ π΅ β π) β (π΄πΊ(1ππ΅)) = (π΄πΊπ΅)) |
| 5 | 4 | fveq2d 6895 | . . . 4 β’ ((π β NrmCVec β§ π΅ β π) β (πβ(π΄πΊ(1ππ΅))) = (πβ(π΄πΊπ΅))) |
| 6 | 5 | oveq1d 7423 | . . 3 β’ ((π β NrmCVec β§ π΅ β π) β ((πβ(π΄πΊ(1ππ΅)))β2) = ((πβ(π΄πΊπ΅))β2)) |
| 7 | 6 | 3adant2 1131 | . 2 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β ((πβ(π΄πΊ(1ππ΅)))β2) = ((πβ(π΄πΊπ΅))β2)) |
| 8 | ax-1cn 11167 | . . 3 β’ 1 β β | |
| 9 | dipfval.2 | . . . 4 β’ πΊ = ( +π£ βπ) | |
| 10 | dipfval.6 | . . . 4 β’ π = (normCVβπ) | |
| 11 | dipfval.7 | . . . 4 β’ π = (Β·πOLDβπ) | |
| 12 | 1, 9, 2, 10, 11 | ipval2lem2 29952 | . . 3 β’ (((π β NrmCVec β§ π΄ β π β§ π΅ β π) β§ 1 β β) β ((πβ(π΄πΊ(1ππ΅)))β2) β β) |
| 13 | 8, 12 | mpan2 689 | . 2 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β ((πβ(π΄πΊ(1ππ΅)))β2) β β) |
| 14 | 7, 13 | eqeltrrd 2834 | 1 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β ((πβ(π΄πΊπ΅))β2) β β) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βcfv 6543 (class class class)co 7408 βcc 11107 βcr 11108 1c1 11110 2c2 12266 βcexp 14026 NrmCVeccnv 29832 +π£ cpv 29833 BaseSetcba 29834 Β·π OLD cns 29835 normCVcnmcv 29838 Β·πOLDcdip 29948 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-n0 12472 df-z 12558 df-uz 12822 df-seq 13966 df-exp 14027 df-grpo 29741 df-ablo 29793 df-vc 29807 df-nv 29840 df-va 29843 df-ba 29844 df-sm 29845 df-0v 29846 df-nmcv 29848 |
| This theorem is referenced by: ipval2 29955 dipcj 29962 dip0r 29965 |
| Copyright terms: Public domain | W3C validator |