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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 | ⊢ (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 2738 | . 2 ⊢ (𝐶 ↾cat 𝐻) = (𝐶 ↾cat 𝐻) | |
2 | rngcrescrhm.c | . . . 4 ⊢ 𝐶 = (RngCat‘𝑈) | |
3 | 2 | fvexi 6682 | . . 3 ⊢ 𝐶 ∈ V |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → 𝐶 ∈ V) |
5 | rngcrescrhm.r | . . . 4 ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) | |
6 | incom 4089 | . . . 4 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring) | |
7 | 5, 6 | eqtrdi 2789 | . . 3 ⊢ (𝜑 → 𝑅 = (𝑈 ∩ Ring)) |
8 | rngcrescrhm.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
9 | inex1g 5184 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝜑 → (𝑈 ∩ Ring) ∈ V) |
11 | 7, 10 | eqeltrd 2833 | . 2 ⊢ (𝜑 → 𝑅 ∈ V) |
12 | inss1 4117 | . . . . . 6 ⊢ (Ring ∩ 𝑈) ⊆ Ring | |
13 | 5, 12 | eqsstrdi 3929 | . . . . 5 ⊢ (𝜑 → 𝑅 ⊆ Ring) |
14 | xpss12 5534 | . . . . 5 ⊢ ((𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring) → (𝑅 × 𝑅) ⊆ (Ring × Ring)) | |
15 | 13, 13, 14 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝑅 × 𝑅) ⊆ (Ring × Ring)) |
16 | rhmfn 44994 | . . . . 5 ⊢ RingHom Fn (Ring × Ring) | |
17 | fnssresb 6452 | . . . . 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 260 | . . 3 ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅)) |
20 | rngcrescrhm.h | . . . 4 ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) | |
21 | 20 | fneq1i 6429 | . . 3 ⊢ (𝐻 Fn (𝑅 × 𝑅) ↔ ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅)) |
22 | 19, 21 | sylibr 237 | . 2 ⊢ (𝜑 → 𝐻 Fn (𝑅 × 𝑅)) |
23 | 1, 4, 11, 22 | rescval2 17196 | 1 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s 𝑅) sSet 〈(Hom ‘ndx), 𝐻〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1542 ∈ wcel 2113 Vcvv 3397 ∩ cin 3840 ⊆ wss 3841 〈cop 4519 × cxp 5517 ↾ cres 5521 Fn wfn 6328 ‘cfv 6333 (class class class)co 7164 ndxcnx 16576 sSet csts 16577 ↾s cress 16580 Hom chom 16672 ↾cat cresc 17176 Ringcrg 19409 RingHom crh 19579 RngCatcrngc 45033 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-er 8313 df-map 8432 df-en 8549 df-dom 8550 df-sdom 8551 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-plusg 16674 df-0g 16811 df-resc 17179 df-mhm 18065 df-ghm 18467 df-mgp 19352 df-ur 19364 df-ring 19411 df-rnghom 19582 |
This theorem is referenced by: (None) |
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