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| Mirrors > Home > MPE Home > Th. List > 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 | ⊢ (𝜑 → 𝑈 ∈ 𝑉) | 
| rngcrescrhm.c | ⊢ 𝐶 = (RngCat‘𝑈) | 
| rngcrescrhm.r | ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) | 
| rngcrescrhm.h | ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) | 
| Ref | Expression | 
|---|---|
| rngcrescrhm | ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s 𝑅) sSet 〈(Hom ‘ndx), 𝐻〉)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2736 | . 2 ⊢ (𝐶 ↾cat 𝐻) = (𝐶 ↾cat 𝐻) | |
| 2 | rngcrescrhm.c | . . . 4 ⊢ 𝐶 = (RngCat‘𝑈) | |
| 3 | 2 | fvexi 6919 | . . 3 ⊢ 𝐶 ∈ V | 
| 4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → 𝐶 ∈ V) | 
| 5 | rngcrescrhm.r | . . . 4 ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) | |
| 6 | incom 4208 | . . . 4 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring) | |
| 7 | 5, 6 | eqtrdi 2792 | . . 3 ⊢ (𝜑 → 𝑅 = (𝑈 ∩ Ring)) | 
| 8 | rngcrescrhm.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
| 9 | inex1g 5318 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V) | |
| 10 | 8, 9 | syl 17 | . . 3 ⊢ (𝜑 → (𝑈 ∩ Ring) ∈ V) | 
| 11 | 7, 10 | eqeltrd 2840 | . 2 ⊢ (𝜑 → 𝑅 ∈ V) | 
| 12 | inss1 4236 | . . . . . 6 ⊢ (Ring ∩ 𝑈) ⊆ Ring | |
| 13 | 5, 12 | eqsstrdi 4027 | . . . . 5 ⊢ (𝜑 → 𝑅 ⊆ Ring) | 
| 14 | xpss12 5699 | . . . . 5 ⊢ ((𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring) → (𝑅 × 𝑅) ⊆ (Ring × Ring)) | |
| 15 | 13, 13, 14 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑅 × 𝑅) ⊆ (Ring × Ring)) | 
| 16 | rhmfn 20500 | . . . . 5 ⊢ RingHom Fn (Ring × Ring) | |
| 17 | fnssresb 6689 | . . . . 5 ⊢ ( RingHom Fn (Ring × Ring) → (( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅) ↔ (𝑅 × 𝑅) ⊆ (Ring × Ring))) | |
| 18 | 16, 17 | mp1i 13 | . . . 4 ⊢ (𝜑 → (( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅) ↔ (𝑅 × 𝑅) ⊆ (Ring × Ring))) | 
| 19 | 15, 18 | mpbird 257 | . . 3 ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅)) | 
| 20 | rngcrescrhm.h | . . . 4 ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) | |
| 21 | 20 | fneq1i 6664 | . . 3 ⊢ (𝐻 Fn (𝑅 × 𝑅) ↔ ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅)) | 
| 22 | 19, 21 | sylibr 234 | . 2 ⊢ (𝜑 → 𝐻 Fn (𝑅 × 𝑅)) | 
| 23 | 1, 4, 11, 22 | rescval2 17873 | 1 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s 𝑅) sSet 〈(Hom ‘ndx), 𝐻〉)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∩ cin 3949 ⊆ wss 3950 〈cop 4631 × cxp 5682 ↾ cres 5686 Fn wfn 6555 ‘cfv 6560 (class class class)co 7432 sSet csts 17201 ndxcnx 17231 ↾s cress 17275 Hom chom 17309 ↾cat cresc 17853 Ringcrg 20231 RingHom crh 20470 RngCatcrngc 20617 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-plusg 17311 df-0g 17487 df-resc 17856 df-mhm 18797 df-ghm 19232 df-mgp 20139 df-ur 20180 df-ring 20233 df-rhm 20473 | 
| This theorem is referenced by: (None) | 
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