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Mirrors > Home > MPE Home > Th. List > estrccofval | Structured version Visualization version GIF version |
Description: Composition in the category of extensible structures. (Contributed by AV, 7-Mar-2020.) |
Ref | Expression |
---|---|
estrcbas.c | β’ πΆ = (ExtStrCatβπ) |
estrcbas.u | β’ (π β π β π) |
estrcco.o | β’ Β· = (compβπΆ) |
Ref | Expression |
---|---|
estrccofval | β’ (π β Β· = (π£ β (π Γ π), π§ β π β¦ (π β ((Baseβπ§) βm (Baseβ(2nd βπ£))), π β ((Baseβ(2nd βπ£)) βm (Baseβ(1st βπ£))) β¦ (π β π)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | estrcbas.c | . . 3 β’ πΆ = (ExtStrCatβπ) | |
2 | estrcbas.u | . . 3 β’ (π β π β π) | |
3 | eqid 2733 | . . . 4 β’ (Hom βπΆ) = (Hom βπΆ) | |
4 | 1, 2, 3 | estrchomfval 18077 | . . 3 β’ (π β (Hom βπΆ) = (π₯ β π, π¦ β π β¦ ((Baseβπ¦) βm (Baseβπ₯)))) |
5 | eqidd 2734 | . . 3 β’ (π β (π£ β (π Γ π), π§ β π β¦ (π β ((Baseβπ§) βm (Baseβ(2nd βπ£))), π β ((Baseβ(2nd βπ£)) βm (Baseβ(1st βπ£))) β¦ (π β π))) = (π£ β (π Γ π), π§ β π β¦ (π β ((Baseβπ§) βm (Baseβ(2nd βπ£))), π β ((Baseβ(2nd βπ£)) βm (Baseβ(1st βπ£))) β¦ (π β π)))) | |
6 | 1, 2, 4, 5 | estrcval 18075 | . 2 β’ (π β πΆ = {β¨(Baseβndx), πβ©, β¨(Hom βndx), (Hom βπΆ)β©, β¨(compβndx), (π£ β (π Γ π), π§ β π β¦ (π β ((Baseβπ§) βm (Baseβ(2nd βπ£))), π β ((Baseβ(2nd βπ£)) βm (Baseβ(1st βπ£))) β¦ (π β π)))β©}) |
7 | catstr 17909 | . 2 β’ {β¨(Baseβndx), πβ©, β¨(Hom βndx), (Hom βπΆ)β©, β¨(compβndx), (π£ β (π Γ π), π§ β π β¦ (π β ((Baseβπ§) βm (Baseβ(2nd βπ£))), π β ((Baseβ(2nd βπ£)) βm (Baseβ(1st βπ£))) β¦ (π β π)))β©} Struct β¨1, ;15β© | |
8 | ccoid 17359 | . 2 β’ comp = Slot (compβndx) | |
9 | snsstp3 4822 | . 2 β’ {β¨(compβndx), (π£ β (π Γ π), π§ β π β¦ (π β ((Baseβπ§) βm (Baseβ(2nd βπ£))), π β ((Baseβ(2nd βπ£)) βm (Baseβ(1st βπ£))) β¦ (π β π)))β©} β {β¨(Baseβndx), πβ©, β¨(Hom βndx), (Hom βπΆ)β©, β¨(compβndx), (π£ β (π Γ π), π§ β π β¦ (π β ((Baseβπ§) βm (Baseβ(2nd βπ£))), π β ((Baseβ(2nd βπ£)) βm (Baseβ(1st βπ£))) β¦ (π β π)))β©} | |
10 | 2, 2 | xpexd 7738 | . . 3 β’ (π β (π Γ π) β V) |
11 | mpoexga 8064 | . . 3 β’ (((π Γ π) β V β§ π β π) β (π£ β (π Γ π), π§ β π β¦ (π β ((Baseβπ§) βm (Baseβ(2nd βπ£))), π β ((Baseβ(2nd βπ£)) βm (Baseβ(1st βπ£))) β¦ (π β π))) β V) | |
12 | 10, 2, 11 | syl2anc 585 | . 2 β’ (π β (π£ β (π Γ π), π§ β π β¦ (π β ((Baseβπ§) βm (Baseβ(2nd βπ£))), π β ((Baseβ(2nd βπ£)) βm (Baseβ(1st βπ£))) β¦ (π β π))) β V) |
13 | estrcco.o | . 2 β’ Β· = (compβπΆ) | |
14 | 6, 7, 8, 9, 12, 13 | strfv3 17138 | 1 β’ (π β Β· = (π£ β (π Γ π), π§ β π β¦ (π β ((Baseβπ§) βm (Baseβ(2nd βπ£))), π β ((Baseβ(2nd βπ£)) βm (Baseβ(1st βπ£))) β¦ (π β π)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3475 {ctp 4633 β¨cop 4635 Γ cxp 5675 β ccom 5681 βcfv 6544 (class class class)co 7409 β cmpo 7411 1st c1st 7973 2nd c2nd 7974 βm cmap 8820 1c1 11111 5c5 12270 ;cdc 12677 ndxcnx 17126 Basecbs 17144 Hom chom 17208 compcco 17209 ExtStrCatcestrc 18073 |
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-tp 4634 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-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 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-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-fz 13485 df-struct 17080 df-slot 17115 df-ndx 17127 df-base 17145 df-hom 17221 df-cco 17222 df-estrc 18074 |
This theorem is referenced by: estrcco 18081 dfrngc2 46870 dfringc2 46916 |
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