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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngchomrnghmresALTV | Structured version Visualization version GIF version |
Description: The value of the functionalized Hom-set operation in the category of non-unital rings (in a universe) as restriction of the non-unital ring homomorphisms. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rngchomrnghmresALTV.c | β’ πΆ = (RngCatALTVβπ) |
rngchomrnghmresALTV.b | β’ π΅ = (Rng β© π) |
rngchomrnghmresALTV.u | β’ (π β π β π) |
rngchomrnghmresALTV.f | β’ πΉ = (Homf βπΆ) |
Ref | Expression |
---|---|
rngchomrnghmresALTV | β’ (π β πΉ = ( RngHom βΎ (π΅ Γ π΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngchomrnghmresALTV.c | . . . . 5 β’ πΆ = (RngCatALTVβπ) | |
2 | eqid 2727 | . . . . 5 β’ (BaseβπΆ) = (BaseβπΆ) | |
3 | rngchomrnghmresALTV.u | . . . . 5 β’ (π β π β π) | |
4 | 1, 2, 3 | rngcbasALTV 47251 | . . . 4 β’ (π β (BaseβπΆ) = (π β© Rng)) |
5 | inss2 4225 | . . . 4 β’ (π β© Rng) β Rng | |
6 | 4, 5 | eqsstrdi 4032 | . . 3 β’ (π β (BaseβπΆ) β Rng) |
7 | resmpo 7534 | . . 3 β’ (((BaseβπΆ) β Rng β§ (BaseβπΆ) β Rng) β ((π₯ β Rng, π¦ β Rng β¦ (π₯ RngHom π¦)) βΎ ((BaseβπΆ) Γ (BaseβπΆ))) = (π₯ β (BaseβπΆ), π¦ β (BaseβπΆ) β¦ (π₯ RngHom π¦))) | |
8 | 6, 6, 7 | syl2anc 583 | . 2 β’ (π β ((π₯ β Rng, π¦ β Rng β¦ (π₯ RngHom π¦)) βΎ ((BaseβπΆ) Γ (BaseβπΆ))) = (π₯ β (BaseβπΆ), π¦ β (BaseβπΆ) β¦ (π₯ RngHom π¦))) |
9 | df-rnghm 20364 | . . . . . 6 β’ RngHom = (π β Rng, π β Rng β¦ β¦(Baseβπ) / π£β¦β¦(Baseβπ ) / π€β¦{π β (π€ βm π£) β£ βπ₯ β π£ βπ¦ β π£ ((πβ(π₯(+gβπ)π¦)) = ((πβπ₯)(+gβπ )(πβπ¦)) β§ (πβ(π₯(.rβπ)π¦)) = ((πβπ₯)(.rβπ )(πβπ¦)))}) | |
10 | ovex 7447 | . . . . . . . . 9 β’ (π€ βm π£) β V | |
11 | 10 | rabex 5328 | . . . . . . . 8 β’ {π β (π€ βm π£) β£ βπ₯ β π£ βπ¦ β π£ ((πβ(π₯(+gβπ)π¦)) = ((πβπ₯)(+gβπ )(πβπ¦)) β§ (πβ(π₯(.rβπ)π¦)) = ((πβπ₯)(.rβπ )(πβπ¦)))} β V |
12 | 11 | csbex 5305 | . . . . . . 7 β’ β¦(Baseβπ ) / π€β¦{π β (π€ βm π£) β£ βπ₯ β π£ βπ¦ β π£ ((πβ(π₯(+gβπ)π¦)) = ((πβπ₯)(+gβπ )(πβπ¦)) β§ (πβ(π₯(.rβπ)π¦)) = ((πβπ₯)(.rβπ )(πβπ¦)))} β V |
13 | 12 | csbex 5305 | . . . . . 6 β’ β¦(Baseβπ) / π£β¦β¦(Baseβπ ) / π€β¦{π β (π€ βm π£) β£ βπ₯ β π£ βπ¦ β π£ ((πβ(π₯(+gβπ)π¦)) = ((πβπ₯)(+gβπ )(πβπ¦)) β§ (πβ(π₯(.rβπ)π¦)) = ((πβπ₯)(.rβπ )(πβπ¦)))} β V |
14 | 9, 13 | fnmpoi 8068 | . . . . 5 β’ RngHom Fn (Rng Γ Rng) |
15 | 14 | a1i 11 | . . . 4 β’ (π β RngHom Fn (Rng Γ Rng)) |
16 | fnov 7546 | . . . 4 β’ ( RngHom Fn (Rng Γ Rng) β RngHom = (π₯ β Rng, π¦ β Rng β¦ (π₯ RngHom π¦))) | |
17 | 15, 16 | sylib 217 | . . 3 β’ (π β RngHom = (π₯ β Rng, π¦ β Rng β¦ (π₯ RngHom π¦))) |
18 | incom 4197 | . . . . . 6 β’ (π β© Rng) = (Rng β© π) | |
19 | 18 | a1i 11 | . . . . 5 β’ (π β (π β© Rng) = (Rng β© π)) |
20 | rngchomrnghmresALTV.b | . . . . . 6 β’ π΅ = (Rng β© π) | |
21 | 20 | a1i 11 | . . . . 5 β’ (π β π΅ = (Rng β© π)) |
22 | 19, 4, 21 | 3eqtr4rd 2778 | . . . 4 β’ (π β π΅ = (BaseβπΆ)) |
23 | 22 | sqxpeqd 5704 | . . 3 β’ (π β (π΅ Γ π΅) = ((BaseβπΆ) Γ (BaseβπΆ))) |
24 | 17, 23 | reseq12d 5980 | . 2 β’ (π β ( RngHom βΎ (π΅ Γ π΅)) = ((π₯ β Rng, π¦ β Rng β¦ (π₯ RngHom π¦)) βΎ ((BaseβπΆ) Γ (BaseβπΆ)))) |
25 | rngchomrnghmresALTV.f | . . 3 β’ πΉ = (Homf βπΆ) | |
26 | 1, 2, 3, 25 | rngchomffvalALTV 47263 | . 2 β’ (π β πΉ = (π₯ β (BaseβπΆ), π¦ β (BaseβπΆ) β¦ (π₯ RngHom π¦))) |
27 | 8, 24, 26 | 3eqtr4rd 2778 | 1 β’ (π β πΉ = ( RngHom βΎ (π΅ Γ π΅))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 βwral 3056 {crab 3427 β¦csb 3889 β© cin 3943 β wss 3944 Γ cxp 5670 βΎ cres 5674 Fn wfn 6537 βcfv 6542 (class class class)co 7414 β cmpo 7416 βm cmap 8836 Basecbs 17171 +gcplusg 17224 .rcmulr 17225 Homf chomf 17637 Rngcrng 20083 RngHom crnghm 20362 RngCatALTVcrngcALTV 47248 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-fz 13509 df-struct 17107 df-slot 17142 df-ndx 17154 df-base 17172 df-hom 17248 df-cco 17249 df-homf 17641 df-rnghm 20364 df-rngcALTV 47249 |
This theorem is referenced by: rhmsubcALTV 47270 |
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