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Mirrors > Home > MPE Home > Th. List > opsrbaslem | Structured version Visualization version GIF version |
Description: 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.) (Revised by AV, 1-Nov-2024.) |
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)) |
opsrbaslem.1 | β’ πΈ = Slot (πΈβndx) |
opsrbaslem.2 | β’ (πΈβndx) β (leβndx) |
Ref | Expression |
---|---|
opsrbaslem | β’ (π β (πΈβπ) = (πΈβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opsrbaslem.1 | . . . 4 β’ πΈ = Slot (πΈβndx) | |
2 | opsrbaslem.2 | . . . 4 β’ (πΈβndx) β (leβndx) | |
3 | 1, 2 | setsnid 17016 | . . 3 β’ (πΈβπ) = (πΈβ(π sSet β¨(leβndx), (leβπ)β©)) |
4 | opsrbas.s | . . . . 5 β’ π = (πΌ mPwSer π ) | |
5 | opsrbas.o | . . . . 5 β’ π = ((πΌ ordPwSer π )βπ) | |
6 | eqid 2738 | . . . . 5 β’ (leβπ) = (leβπ) | |
7 | simprl 770 | . . . . 5 β’ ((π β§ (πΌ β V β§ π β V)) β πΌ β V) | |
8 | simprr 772 | . . . . 5 β’ ((π β§ (πΌ β V β§ π β V)) β π β V) | |
9 | opsrbas.t | . . . . . 6 β’ (π β π β (πΌ Γ πΌ)) | |
10 | 9 | adantr 482 | . . . . 5 β’ ((π β§ (πΌ β V β§ π β V)) β π β (πΌ Γ πΌ)) |
11 | 4, 5, 6, 7, 8, 10 | opsrval2 21371 | . . . 4 β’ ((π β§ (πΌ β V β§ π β V)) β π = (π sSet β¨(leβndx), (leβπ)β©)) |
12 | 11 | fveq2d 6842 | . . 3 β’ ((π β§ (πΌ β V β§ π β V)) β (πΈβπ) = (πΈβ(π sSet β¨(leβndx), (leβπ)β©))) |
13 | 3, 12 | eqtr4id 2797 | . 2 β’ ((π β§ (πΌ β V β§ π β V)) β (πΈβπ) = (πΈβπ)) |
14 | 0fv 6882 | . . . . . . 7 β’ (β βπ) = β | |
15 | 14 | eqcomi 2747 | . . . . . 6 β’ β = (β βπ) |
16 | reldmpsr 21239 | . . . . . . 7 β’ Rel dom mPwSer | |
17 | 16 | ovprc 7388 | . . . . . 6 β’ (Β¬ (πΌ β V β§ π β V) β (πΌ mPwSer π ) = β ) |
18 | reldmopsr 21368 | . . . . . . . 8 β’ Rel dom ordPwSer | |
19 | 18 | ovprc 7388 | . . . . . . 7 β’ (Β¬ (πΌ β V β§ π β V) β (πΌ ordPwSer π ) = β ) |
20 | 19 | fveq1d 6840 | . . . . . 6 β’ (Β¬ (πΌ β V β§ π β V) β ((πΌ ordPwSer π )βπ) = (β βπ)) |
21 | 15, 17, 20 | 3eqtr4a 2804 | . . . . 5 β’ (Β¬ (πΌ β V β§ π β V) β (πΌ mPwSer π ) = ((πΌ ordPwSer π )βπ)) |
22 | 21, 4, 5 | 3eqtr4g 2803 | . . . 4 β’ (Β¬ (πΌ β V β§ π β V) β π = π) |
23 | 22 | fveq2d 6842 | . . 3 β’ (Β¬ (πΌ β V β§ π β V) β (πΈβπ) = (πΈβπ)) |
24 | 23 | adantl 483 | . 2 β’ ((π β§ Β¬ (πΌ β V β§ π β V)) β (πΈβπ) = (πΈβπ)) |
25 | 13, 24 | 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 2942 Vcvv 3444 β wss 3909 β c0 4281 β¨cop 4591 Γ cxp 5629 βcfv 6492 (class class class)co 7350 sSet csts 16970 Slot cslot 16988 ndxcnx 17000 lecple 17075 mPwSer cmps 21229 ordPwSer copws 21233 |
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 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-pnf 11125 df-mnf 11126 df-ltxr 11128 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-7 12155 df-8 12156 df-9 12157 df-dec 12552 df-sets 16971 df-slot 16989 df-ndx 17001 df-base 17019 df-ple 17088 df-psr 21234 df-opsr 21238 |
This theorem is referenced by: opsrbas 21374 opsrplusg 21376 opsrmulr 21378 opsrvsca 21380 opsrsca 21382 |
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