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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngchomfval | Structured version Visualization version GIF version |
Description: Set of arrows of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
Ref | Expression |
---|---|
rngcbas.c | โข ๐ถ = (RngCatโ๐) |
rngcbas.b | โข ๐ต = (Baseโ๐ถ) |
rngcbas.u | โข (๐ โ ๐ โ ๐) |
rngchomfval.h | โข ๐ป = (Hom โ๐ถ) |
Ref | Expression |
---|---|
rngchomfval | โข (๐ โ ๐ป = ( RngHomo โพ (๐ต ร ๐ต))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngchomfval.h | . . 3 โข ๐ป = (Hom โ๐ถ) | |
2 | rngcbas.c | . . . . 5 โข ๐ถ = (RngCatโ๐) | |
3 | rngcbas.u | . . . . 5 โข (๐ โ ๐ โ ๐) | |
4 | rngcbas.b | . . . . . 6 โข ๐ต = (Baseโ๐ถ) | |
5 | 2, 4, 3 | rngcbas 46948 | . . . . 5 โข (๐ โ ๐ต = (๐ โฉ Rng)) |
6 | eqidd 2733 | . . . . 5 โข (๐ โ ( RngHomo โพ (๐ต ร ๐ต)) = ( RngHomo โพ (๐ต ร ๐ต))) | |
7 | 2, 3, 5, 6 | rngcval 46945 | . . . 4 โข (๐ โ ๐ถ = ((ExtStrCatโ๐) โพcat ( RngHomo โพ (๐ต ร ๐ต)))) |
8 | 7 | fveq2d 6895 | . . 3 โข (๐ โ (Hom โ๐ถ) = (Hom โ((ExtStrCatโ๐) โพcat ( RngHomo โพ (๐ต ร ๐ต))))) |
9 | 1, 8 | eqtrid 2784 | . 2 โข (๐ โ ๐ป = (Hom โ((ExtStrCatโ๐) โพcat ( RngHomo โพ (๐ต ร ๐ต))))) |
10 | eqid 2732 | . . 3 โข ((ExtStrCatโ๐) โพcat ( RngHomo โพ (๐ต ร ๐ต))) = ((ExtStrCatโ๐) โพcat ( RngHomo โพ (๐ต ร ๐ต))) | |
11 | eqid 2732 | . . 3 โข (Baseโ(ExtStrCatโ๐)) = (Baseโ(ExtStrCatโ๐)) | |
12 | fvexd 6906 | . . 3 โข (๐ โ (ExtStrCatโ๐) โ V) | |
13 | 5, 6 | rnghmresfn 46946 | . . 3 โข (๐ โ ( RngHomo โพ (๐ต ร ๐ต)) Fn (๐ต ร ๐ต)) |
14 | inss1 4228 | . . . . 5 โข (๐ โฉ Rng) โ ๐ | |
15 | 14 | a1i 11 | . . . 4 โข (๐ โ (๐ โฉ Rng) โ ๐) |
16 | eqid 2732 | . . . . . 6 โข (ExtStrCatโ๐) = (ExtStrCatโ๐) | |
17 | 16, 3 | estrcbas 18078 | . . . . 5 โข (๐ โ ๐ = (Baseโ(ExtStrCatโ๐))) |
18 | 17 | eqcomd 2738 | . . . 4 โข (๐ โ (Baseโ(ExtStrCatโ๐)) = ๐) |
19 | 15, 5, 18 | 3sstr4d 4029 | . . 3 โข (๐ โ ๐ต โ (Baseโ(ExtStrCatโ๐))) |
20 | 10, 11, 12, 13, 19 | reschom 17780 | . 2 โข (๐ โ ( RngHomo โพ (๐ต ร ๐ต)) = (Hom โ((ExtStrCatโ๐) โพcat ( RngHomo โพ (๐ต ร ๐ต))))) |
21 | 9, 20 | eqtr4d 2775 | 1 โข (๐ โ ๐ป = ( RngHomo โพ (๐ต ร ๐ต))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1541 โ wcel 2106 Vcvv 3474 โฉ cin 3947 โ wss 3948 ร cxp 5674 โพ cres 5678 โcfv 6543 (class class class)co 7411 Basecbs 17146 Hom chom 17210 โพcat cresc 17757 ExtStrCatcestrc 18075 Rngcrng 46733 RngHomo crngh 46768 RngCatcrngc 46940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 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-tp 4633 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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-fz 13487 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-hom 17223 df-cco 17224 df-resc 17760 df-estrc 18076 df-rnghomo 46770 df-rngc 46942 |
This theorem is referenced by: rngchom 46950 rngchomfeqhom 46952 rngccofval 46953 rnghmsubcsetclem1 46958 rngcifuestrc 46980 funcrngcsetc 46981 rhmsubcrngc 47012 rhmsubc 47073 |
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