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Mirrors > Home > MPE Home > Th. List > opsrval2 | Structured version Visualization version GIF version |
Description: Self-referential expression for the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) |
Ref | Expression |
---|---|
opsrval2.s | β’ π = (πΌ mPwSer π ) |
opsrval2.o | β’ π = ((πΌ ordPwSer π )βπ) |
opsrval2.l | β’ β€ = (leβπ) |
opsrval2.i | β’ (π β πΌ β π) |
opsrval2.r | β’ (π β π β π) |
opsrval2.t | β’ (π β π β (πΌ Γ πΌ)) |
Ref | Expression |
---|---|
opsrval2 | β’ (π β π = (π sSet β¨(leβndx), β€ β©)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opsrval2.s | . . 3 β’ π = (πΌ mPwSer π ) | |
2 | opsrval2.o | . . 3 β’ π = ((πΌ ordPwSer π )βπ) | |
3 | eqid 2733 | . . 3 β’ (Baseβπ) = (Baseβπ) | |
4 | eqid 2733 | . . 3 β’ (ltβπ ) = (ltβπ ) | |
5 | eqid 2733 | . . 3 β’ (π <bag πΌ) = (π <bag πΌ) | |
6 | eqid 2733 | . . 3 β’ {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} = {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} | |
7 | eqid 2733 | . . 3 β’ {β¨π₯, π¦β© β£ ({π₯, π¦} β (Baseβπ) β§ (βπ§ β {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} ((π₯βπ§)(ltβπ )(π¦βπ§) β§ βπ€ β {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} (π€(π <bag πΌ)π§ β (π₯βπ€) = (π¦βπ€))) β¨ π₯ = π¦))} = {β¨π₯, π¦β© β£ ({π₯, π¦} β (Baseβπ) β§ (βπ§ β {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} ((π₯βπ§)(ltβπ )(π¦βπ§) β§ βπ€ β {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} (π€(π <bag πΌ)π§ β (π₯βπ€) = (π¦βπ€))) β¨ π₯ = π¦))} | |
8 | opsrval2.i | . . 3 β’ (π β πΌ β π) | |
9 | opsrval2.r | . . 3 β’ (π β π β π) | |
10 | opsrval2.t | . . 3 β’ (π β π β (πΌ Γ πΌ)) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | opsrval 21463 | . 2 β’ (π β π = (π sSet β¨(leβndx), {β¨π₯, π¦β© β£ ({π₯, π¦} β (Baseβπ) β§ (βπ§ β {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} ((π₯βπ§)(ltβπ )(π¦βπ§) β§ βπ€ β {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} (π€(π <bag πΌ)π§ β (π₯βπ€) = (π¦βπ€))) β¨ π₯ = π¦))}β©)) |
12 | opsrval2.l | . . . . 5 β’ β€ = (leβπ) | |
13 | 1, 2, 3, 4, 5, 6, 12, 10 | opsrle 21464 | . . . 4 β’ (π β β€ = {β¨π₯, π¦β© β£ ({π₯, π¦} β (Baseβπ) β§ (βπ§ β {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} ((π₯βπ§)(ltβπ )(π¦βπ§) β§ βπ€ β {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} (π€(π <bag πΌ)π§ β (π₯βπ€) = (π¦βπ€))) β¨ π₯ = π¦))}) |
14 | 13 | opeq2d 4838 | . . 3 β’ (π β β¨(leβndx), β€ β© = β¨(leβndx), {β¨π₯, π¦β© β£ ({π₯, π¦} β (Baseβπ) β§ (βπ§ β {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} ((π₯βπ§)(ltβπ )(π¦βπ§) β§ βπ€ β {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} (π€(π <bag πΌ)π§ β (π₯βπ€) = (π¦βπ€))) β¨ π₯ = π¦))}β©) |
15 | 14 | oveq2d 7374 | . 2 β’ (π β (π sSet β¨(leβndx), β€ β©) = (π sSet β¨(leβndx), {β¨π₯, π¦β© β£ ({π₯, π¦} β (Baseβπ) β§ (βπ§ β {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} ((π₯βπ§)(ltβπ )(π¦βπ§) β§ βπ€ β {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} (π€(π <bag πΌ)π§ β (π₯βπ€) = (π¦βπ€))) β¨ π₯ = π¦))}β©)) |
16 | 11, 15 | eqtr4d 2776 | 1 β’ (π β π = (π sSet β¨(leβndx), β€ β©)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β¨ wo 846 = wceq 1542 β wcel 2107 βwral 3061 βwrex 3070 {crab 3406 β wss 3911 {cpr 4589 β¨cop 4593 class class class wbr 5106 {copab 5168 Γ cxp 5632 β‘ccnv 5633 β cima 5637 βcfv 6497 (class class class)co 7358 βm cmap 8768 Fincfn 8886 βcn 12158 β0cn0 12418 sSet csts 17040 ndxcnx 17070 Basecbs 17088 lecple 17145 ltcplt 18202 mPwSer cmps 21322 <bag cltb 21325 ordPwSer copws 21326 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 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 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-ltxr 11199 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-dec 12624 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ple 17158 df-psr 21327 df-opsr 21331 |
This theorem is referenced by: opsrbaslem 21466 opsrbaslemOLD 21467 |
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