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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngcrescrhm | Structured version Visualization version GIF version |
Description: The category of non-unital rings (in a universe) restricted to the ring homomorphisms between unital rings (in the same universe). (Contributed by AV, 1-Mar-2020.) |
Ref | Expression |
---|---|
rngcrescrhm.u | β’ (π β π β π) |
rngcrescrhm.c | β’ πΆ = (RngCatβπ) |
rngcrescrhm.r | β’ (π β π = (Ring β© π)) |
rngcrescrhm.h | β’ π» = ( RingHom βΎ (π Γ π )) |
Ref | Expression |
---|---|
rngcrescrhm | β’ (π β (πΆ βΎcat π») = ((πΆ βΎs π ) sSet β¨(Hom βndx), π»β©)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . 2 β’ (πΆ βΎcat π») = (πΆ βΎcat π») | |
2 | rngcrescrhm.c | . . . 4 β’ πΆ = (RngCatβπ) | |
3 | 2 | fvexi 6905 | . . 3 β’ πΆ β V |
4 | 3 | a1i 11 | . 2 β’ (π β πΆ β V) |
5 | rngcrescrhm.r | . . . 4 β’ (π β π = (Ring β© π)) | |
6 | incom 4201 | . . . 4 β’ (Ring β© π) = (π β© Ring) | |
7 | 5, 6 | eqtrdi 2787 | . . 3 β’ (π β π = (π β© Ring)) |
8 | rngcrescrhm.u | . . . 4 β’ (π β π β π) | |
9 | inex1g 5319 | . . . 4 β’ (π β π β (π β© Ring) β V) | |
10 | 8, 9 | syl 17 | . . 3 β’ (π β (π β© Ring) β V) |
11 | 7, 10 | eqeltrd 2832 | . 2 β’ (π β π β V) |
12 | inss1 4228 | . . . . . 6 β’ (Ring β© π) β Ring | |
13 | 5, 12 | eqsstrdi 4036 | . . . . 5 β’ (π β π β Ring) |
14 | xpss12 5691 | . . . . 5 β’ ((π β Ring β§ π β Ring) β (π Γ π ) β (Ring Γ Ring)) | |
15 | 13, 13, 14 | syl2anc 583 | . . . 4 β’ (π β (π Γ π ) β (Ring Γ Ring)) |
16 | rhmfn 20391 | . . . . 5 β’ RingHom Fn (Ring Γ Ring) | |
17 | fnssresb 6672 | . . . . 5 β’ ( RingHom Fn (Ring Γ Ring) β (( RingHom βΎ (π Γ π )) Fn (π Γ π ) β (π Γ π ) β (Ring Γ Ring))) | |
18 | 16, 17 | mp1i 13 | . . . 4 β’ (π β (( RingHom βΎ (π Γ π )) Fn (π Γ π ) β (π Γ π ) β (Ring Γ Ring))) |
19 | 15, 18 | mpbird 257 | . . 3 β’ (π β ( RingHom βΎ (π Γ π )) Fn (π Γ π )) |
20 | rngcrescrhm.h | . . . 4 β’ π» = ( RingHom βΎ (π Γ π )) | |
21 | 20 | fneq1i 6646 | . . 3 β’ (π» Fn (π Γ π ) β ( RingHom βΎ (π Γ π )) Fn (π Γ π )) |
22 | 19, 21 | sylibr 233 | . 2 β’ (π β π» Fn (π Γ π )) |
23 | 1, 4, 11, 22 | rescval2 17780 | 1 β’ (π β (πΆ βΎcat π») = ((πΆ βΎs π ) sSet β¨(Hom βndx), π»β©)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1540 β wcel 2105 Vcvv 3473 β© cin 3947 β wss 3948 β¨cop 4634 Γ cxp 5674 βΎ cres 5678 Fn wfn 6538 βcfv 6543 (class class class)co 7412 sSet csts 17101 ndxcnx 17131 βΎs cress 17178 Hom chom 17213 βΎcat cresc 17760 Ringcrg 20128 RingHom crh 20361 RngCatcrngc 46944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-0g 17392 df-resc 17763 df-mhm 18706 df-ghm 19129 df-mgp 20030 df-ur 20077 df-ring 20130 df-rhm 20364 |
This theorem is referenced by: (None) |
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