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 2819 | . 2 ⊢ (𝐶 ↾cat 𝐻) = (𝐶 ↾cat 𝐻) | |
2 | rngcrescrhm.c | . . . 4 ⊢ 𝐶 = (RngCat‘𝑈) | |
3 | 2 | fvexi 6677 | . . 3 ⊢ 𝐶 ∈ V |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → 𝐶 ∈ V) |
5 | rngcrescrhm.r | . . . 4 ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) | |
6 | incom 4176 | . . . 4 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring) | |
7 | 5, 6 | syl6eq 2870 | . . 3 ⊢ (𝜑 → 𝑅 = (𝑈 ∩ Ring)) |
8 | rngcrescrhm.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
9 | inex1g 5214 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝜑 → (𝑈 ∩ Ring) ∈ V) |
11 | 7, 10 | eqeltrd 2911 | . 2 ⊢ (𝜑 → 𝑅 ∈ V) |
12 | inss1 4203 | . . . . . 6 ⊢ (Ring ∩ 𝑈) ⊆ Ring | |
13 | 5, 12 | eqsstrdi 4019 | . . . . 5 ⊢ (𝜑 → 𝑅 ⊆ Ring) |
14 | xpss12 5563 | . . . . 5 ⊢ ((𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring) → (𝑅 × 𝑅) ⊆ (Ring × Ring)) | |
15 | 13, 13, 14 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝑅 × 𝑅) ⊆ (Ring × Ring)) |
16 | rhmfn 44180 | . . . . 5 ⊢ RingHom Fn (Ring × Ring) | |
17 | fnssresb 6462 | . . . . 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 6443 | . . 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 17090 | 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 1531 ∈ wcel 2108 Vcvv 3493 ∩ cin 3933 ⊆ wss 3934 〈cop 4565 × cxp 5546 ↾ cres 5550 Fn wfn 6343 ‘cfv 6348 (class class class)co 7148 ndxcnx 16472 sSet csts 16473 ↾s cress 16476 Hom chom 16568 ↾cat cresc 17070 Ringcrg 19289 RingHom crh 19456 RngCatcrngc 44219 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-map 8400 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-0g 16707 df-resc 17073 df-mhm 17948 df-ghm 18348 df-mgp 19232 df-ur 19244 df-ring 19291 df-rnghom 19459 |
This theorem is referenced by: (None) |
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