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Mirrors > Home > MPE Home > Th. List > ipval2lem2 | Structured version Visualization version GIF version |
Description: Lemma for ipval3 29962. (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 |
---|---|
ipval2lem2 | 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 | simpl1 1192 | . . 3 β’ (((π β NrmCVec β§ π΄ β π β§ π΅ β π) β§ πΆ β β) β π β NrmCVec) | |
2 | simpl2 1193 | . . . 4 β’ (((π β NrmCVec β§ π΄ β π β§ π΅ β π) β§ πΆ β β) β π΄ β π) | |
3 | dipfval.1 | . . . . . . . 8 β’ π = (BaseSetβπ) | |
4 | dipfval.4 | . . . . . . . 8 β’ π = ( Β·π OLD βπ) | |
5 | 3, 4 | nvscl 29879 | . . . . . . 7 β’ ((π β NrmCVec β§ πΆ β β β§ π΅ β π) β (πΆππ΅) β π) |
6 | 5 | 3com23 1127 | . . . . . 6 β’ ((π β NrmCVec β§ π΅ β π β§ πΆ β β) β (πΆππ΅) β π) |
7 | 6 | 3expa 1119 | . . . . 5 β’ (((π β NrmCVec β§ π΅ β π) β§ πΆ β β) β (πΆππ΅) β π) |
8 | 7 | 3adantl2 1168 | . . . 4 β’ (((π β NrmCVec β§ π΄ β π β§ π΅ β π) β§ πΆ β β) β (πΆππ΅) β π) |
9 | dipfval.2 | . . . . 5 β’ πΊ = ( +π£ βπ) | |
10 | 3, 9 | nvgcl 29873 | . . . 4 β’ ((π β NrmCVec β§ π΄ β π β§ (πΆππ΅) β π) β (π΄πΊ(πΆππ΅)) β π) |
11 | 1, 2, 8, 10 | syl3anc 1372 | . . 3 β’ (((π β NrmCVec β§ π΄ β π β§ π΅ β π) β§ πΆ β β) β (π΄πΊ(πΆππ΅)) β π) |
12 | dipfval.6 | . . . 4 β’ π = (normCVβπ) | |
13 | 3, 12 | nvcl 29914 | . . 3 β’ ((π β NrmCVec β§ (π΄πΊ(πΆππ΅)) β π) β (πβ(π΄πΊ(πΆππ΅))) β β) |
14 | 1, 11, 13 | syl2anc 585 | . 2 β’ (((π β NrmCVec β§ π΄ β π β§ π΅ β π) β§ πΆ β β) β (πβ(π΄πΊ(πΆππ΅))) β β) |
15 | 14 | resqcld 14090 | 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 6544 (class class class)co 7409 βcc 11108 βcr 11109 2c2 12267 βcexp 14027 NrmCVeccnv 29837 +π£ cpv 29838 BaseSetcba 29839 Β·π OLD cns 29840 normCVcnmcv 29843 Β·πOLDcdip 29953 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
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 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-n0 12473 df-z 12559 df-uz 12823 df-seq 13967 df-exp 14028 df-grpo 29746 df-ablo 29798 df-vc 29812 df-nv 29845 df-va 29848 df-ba 29849 df-sm 29850 df-0v 29851 df-nmcv 29853 |
This theorem is referenced by: ipval2lem3 29958 ipval2lem4 29959 dipcj 29967 |
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