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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 2735 | . 2 ⊢ (𝐶 ↾cat 𝐻) = (𝐶 ↾cat 𝐻) | |
2 | rngcrescrhm.c | . . . 4 ⊢ 𝐶 = (RngCat‘𝑈) | |
3 | 2 | fvexi 6921 | . . 3 ⊢ 𝐶 ∈ V |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → 𝐶 ∈ V) |
5 | rngcrescrhm.r | . . . 4 ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) | |
6 | incom 4217 | . . . 4 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring) | |
7 | 5, 6 | eqtrdi 2791 | . . 3 ⊢ (𝜑 → 𝑅 = (𝑈 ∩ Ring)) |
8 | rngcrescrhm.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
9 | inex1g 5325 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝜑 → (𝑈 ∩ Ring) ∈ V) |
11 | 7, 10 | eqeltrd 2839 | . 2 ⊢ (𝜑 → 𝑅 ∈ V) |
12 | inss1 4245 | . . . . . 6 ⊢ (Ring ∩ 𝑈) ⊆ Ring | |
13 | 5, 12 | eqsstrdi 4050 | . . . . 5 ⊢ (𝜑 → 𝑅 ⊆ Ring) |
14 | xpss12 5704 | . . . . 5 ⊢ ((𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring) → (𝑅 × 𝑅) ⊆ (Ring × Ring)) | |
15 | 13, 13, 14 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑅 × 𝑅) ⊆ (Ring × Ring)) |
16 | rhmfn 20516 | . . . . 5 ⊢ RingHom Fn (Ring × Ring) | |
17 | fnssresb 6691 | . . . . 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 6666 | . . 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 17876 | 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 1537 ∈ wcel 2106 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 〈cop 4637 × cxp 5687 ↾ cres 5691 Fn wfn 6558 ‘cfv 6563 (class class class)co 7431 sSet csts 17197 ndxcnx 17227 ↾s cress 17274 Hom chom 17309 ↾cat cresc 17856 Ringcrg 20251 RingHom crh 20486 RngCatcrngc 20633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-0g 17488 df-resc 17859 df-mhm 18809 df-ghm 19244 df-mgp 20153 df-ur 20200 df-ring 20253 df-rhm 20489 |
This theorem is referenced by: (None) |
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