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Mirrors > Home > MPE Home > Th. List > opsrbaslemOLD | Structured version Visualization version GIF version |
Description: Obsolete version of opsrbaslem 21453 as of 1-Nov-2024. Get a component of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 9-Sep-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
opsrbas.s | β’ π = (πΌ mPwSer π ) |
opsrbas.o | β’ π = ((πΌ ordPwSer π )βπ) |
opsrbas.t | β’ (π β π β (πΌ Γ πΌ)) |
opsrbaslemOLD.1 | β’ πΈ = Slot π |
opsrbaslemOLD.2 | β’ π β β |
opsrbaslemOLD.3 | β’ π < ;10 |
Ref | Expression |
---|---|
opsrbaslemOLD | β’ (π β (πΈβπ) = (πΈβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opsrbaslemOLD.1 | . . . . 5 β’ πΈ = Slot π | |
2 | opsrbaslemOLD.2 | . . . . 5 β’ π β β | |
3 | 1, 2 | ndxid 17070 | . . . 4 β’ πΈ = Slot (πΈβndx) |
4 | 2 | nnrei 12163 | . . . . . 6 β’ π β β |
5 | opsrbaslemOLD.3 | . . . . . 6 β’ π < ;10 | |
6 | 4, 5 | ltneii 11269 | . . . . 5 β’ π β ;10 |
7 | 1, 2 | ndxarg 17069 | . . . . . 6 β’ (πΈβndx) = π |
8 | plendx 17248 | . . . . . 6 β’ (leβndx) = ;10 | |
9 | 7, 8 | neeq12i 3011 | . . . . 5 β’ ((πΈβndx) β (leβndx) β π β ;10) |
10 | 6, 9 | mpbir 230 | . . . 4 β’ (πΈβndx) β (leβndx) |
11 | 3, 10 | setsnid 17082 | . . 3 β’ (πΈβπ) = (πΈβ(π sSet β¨(leβndx), (leβπ)β©)) |
12 | opsrbas.s | . . . . 5 β’ π = (πΌ mPwSer π ) | |
13 | opsrbas.o | . . . . 5 β’ π = ((πΌ ordPwSer π )βπ) | |
14 | eqid 2737 | . . . . 5 β’ (leβπ) = (leβπ) | |
15 | simprl 770 | . . . . 5 β’ ((π β§ (πΌ β V β§ π β V)) β πΌ β V) | |
16 | simprr 772 | . . . . 5 β’ ((π β§ (πΌ β V β§ π β V)) β π β V) | |
17 | opsrbas.t | . . . . . 6 β’ (π β π β (πΌ Γ πΌ)) | |
18 | 17 | adantr 482 | . . . . 5 β’ ((π β§ (πΌ β V β§ π β V)) β π β (πΌ Γ πΌ)) |
19 | 12, 13, 14, 15, 16, 18 | opsrval2 21452 | . . . 4 β’ ((π β§ (πΌ β V β§ π β V)) β π = (π sSet β¨(leβndx), (leβπ)β©)) |
20 | 19 | fveq2d 6847 | . . 3 β’ ((π β§ (πΌ β V β§ π β V)) β (πΈβπ) = (πΈβ(π sSet β¨(leβndx), (leβπ)β©))) |
21 | 11, 20 | eqtr4id 2796 | . 2 β’ ((π β§ (πΌ β V β§ π β V)) β (πΈβπ) = (πΈβπ)) |
22 | 0fv 6887 | . . . . . . 7 β’ (β βπ) = β | |
23 | 22 | eqcomi 2746 | . . . . . 6 β’ β = (β βπ) |
24 | reldmpsr 21319 | . . . . . . 7 β’ Rel dom mPwSer | |
25 | 24 | ovprc 7396 | . . . . . 6 β’ (Β¬ (πΌ β V β§ π β V) β (πΌ mPwSer π ) = β ) |
26 | reldmopsr 21449 | . . . . . . . 8 β’ Rel dom ordPwSer | |
27 | 26 | ovprc 7396 | . . . . . . 7 β’ (Β¬ (πΌ β V β§ π β V) β (πΌ ordPwSer π ) = β ) |
28 | 27 | fveq1d 6845 | . . . . . 6 β’ (Β¬ (πΌ β V β§ π β V) β ((πΌ ordPwSer π )βπ) = (β βπ)) |
29 | 23, 25, 28 | 3eqtr4a 2803 | . . . . 5 β’ (Β¬ (πΌ β V β§ π β V) β (πΌ mPwSer π ) = ((πΌ ordPwSer π )βπ)) |
30 | 29 | adantl 483 | . . . 4 β’ ((π β§ Β¬ (πΌ β V β§ π β V)) β (πΌ mPwSer π ) = ((πΌ ordPwSer π )βπ)) |
31 | 30, 12, 13 | 3eqtr4g 2802 | . . 3 β’ ((π β§ Β¬ (πΌ β V β§ π β V)) β π = π) |
32 | 31 | fveq2d 6847 | . 2 β’ ((π β§ Β¬ (πΌ β V β§ π β V)) β (πΈβπ) = (πΈβπ)) |
33 | 21, 32 | pm2.61dan 812 | 1 β’ (π β (πΈβπ) = (πΈβπ)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2944 Vcvv 3446 β wss 3911 β c0 4283 β¨cop 4593 class class class wbr 5106 Γ cxp 5632 βcfv 6497 (class class class)co 7358 0cc0 11052 1c1 11053 < clt 11190 βcn 12154 ;cdc 12619 sSet csts 17036 Slot cslot 17054 ndxcnx 17066 lecple 17141 mPwSer cmps 21309 ordPwSer copws 21313 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-ltxr 11195 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 df-8 12223 df-9 12224 df-dec 12620 df-sets 17037 df-slot 17055 df-ndx 17067 df-base 17085 df-ple 17154 df-psr 21314 df-opsr 21318 |
This theorem is referenced by: opsrbasOLD 21456 opsrplusgOLD 21458 opsrmulrOLD 21460 opsrvscaOLD 21462 opsrscaOLD 21464 |
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