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| Mirrors > Home > MPE Home > Th. List > ipval2lem3 | Structured version Visualization version GIF version | ||
| Description: Lemma for ipval3 30563. (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 30481 | . . . . . 6 β’ ((π β NrmCVec β§ π΅ β π) β (1ππ΅) = π΅) |
| 4 | 3 | oveq2d 7432 | . . . . 5 β’ ((π β NrmCVec β§ π΅ β π) β (π΄πΊ(1ππ΅)) = (π΄πΊπ΅)) |
| 5 | 4 | fveq2d 6896 | . . . 4 β’ ((π β NrmCVec β§ π΅ β π) β (πβ(π΄πΊ(1ππ΅))) = (πβ(π΄πΊπ΅))) |
| 6 | 5 | oveq1d 7431 | . . 3 β’ ((π β NrmCVec β§ π΅ β π) β ((πβ(π΄πΊ(1ππ΅)))β2) = ((πβ(π΄πΊπ΅))β2)) |
| 7 | 6 | 3adant2 1128 | . 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 11196 | . . 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 30558 | . . 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 2826 | 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 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6543 (class class class)co 7416 βcc 11136 βcr 11137 1c1 11139 2c2 12297 βcexp 14058 NrmCVeccnv 30438 +π£ cpv 30439 BaseSetcba 30440 Β·π OLD cns 30441 normCVcnmcv 30444 Β·πOLDcdip 30554 |
| 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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 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 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-n0 12503 df-z 12589 df-uz 12853 df-seq 13999 df-exp 14059 df-grpo 30347 df-ablo 30399 df-vc 30413 df-nv 30446 df-va 30449 df-ba 30450 df-sm 30451 df-0v 30452 df-nmcv 30454 |
| This theorem is referenced by: ipval2 30561 dipcj 30568 dip0r 30571 |
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