Mathbox for Alexander van der Vekens |
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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 2821 | . 2 ⊢ (𝐶 ↾cat 𝐻) = (𝐶 ↾cat 𝐻) | |
2 | rngcrescrhm.c | . . . 4 ⊢ 𝐶 = (RngCat‘𝑈) | |
3 | 2 | fvexi 6678 | . . 3 ⊢ 𝐶 ∈ V |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → 𝐶 ∈ V) |
5 | rngcrescrhm.r | . . . 4 ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) | |
6 | incom 4177 | . . . 4 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring) | |
7 | 5, 6 | syl6eq 2872 | . . 3 ⊢ (𝜑 → 𝑅 = (𝑈 ∩ Ring)) |
8 | rngcrescrhm.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
9 | inex1g 5215 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝜑 → (𝑈 ∩ Ring) ∈ V) |
11 | 7, 10 | eqeltrd 2913 | . 2 ⊢ (𝜑 → 𝑅 ∈ V) |
12 | inss1 4204 | . . . . . 6 ⊢ (Ring ∩ 𝑈) ⊆ Ring | |
13 | 5, 12 | eqsstrdi 4020 | . . . . 5 ⊢ (𝜑 → 𝑅 ⊆ Ring) |
14 | xpss12 5564 | . . . . 5 ⊢ ((𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring) → (𝑅 × 𝑅) ⊆ (Ring × Ring)) | |
15 | 13, 13, 14 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝑅 × 𝑅) ⊆ (Ring × Ring)) |
16 | rhmfn 44183 | . . . . 5 ⊢ RingHom Fn (Ring × Ring) | |
17 | fnssresb 6463 | . . . . 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 259 | . . 3 ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅)) |
20 | rngcrescrhm.h | . . . 4 ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) | |
21 | 20 | fneq1i 6444 | . . 3 ⊢ (𝐻 Fn (𝑅 × 𝑅) ↔ ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅)) |
22 | 19, 21 | sylibr 236 | . 2 ⊢ (𝜑 → 𝐻 Fn (𝑅 × 𝑅)) |
23 | 1, 4, 11, 22 | rescval2 17092 | 1 ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s 𝑅) sSet 〈(Hom ‘ndx), 𝐻〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∩ cin 3934 ⊆ wss 3935 〈cop 4566 × cxp 5547 ↾ cres 5551 Fn wfn 6344 ‘cfv 6349 (class class class)co 7150 ndxcnx 16474 sSet csts 16475 ↾s cress 16478 Hom chom 16570 ↾cat cresc 17072 Ringcrg 19291 RingHom crh 19458 RngCatcrngc 44222 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-plusg 16572 df-0g 16709 df-resc 17075 df-mhm 17950 df-ghm 18350 df-mgp 19234 df-ur 19246 df-ring 19293 df-rnghom 19461 |
This theorem is referenced by: (None) |
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