| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > dfringc2 | Structured version Visualization version GIF version | ||
| Description: Alternate definition of the category of unital rings (in a universe). (Contributed by AV, 16-Mar-2020.) |
| Ref | Expression |
|---|---|
| dfringc2.c | ⊢ 𝐶 = (RingCat‘𝑈) |
| dfringc2.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
| dfringc2.b | ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) |
| dfringc2.h | ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) |
| dfringc2.o | ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈))) |
| Ref | Expression |
|---|---|
| dfringc2 | ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfringc2.c | . . 3 ⊢ 𝐶 = (RingCat‘𝑈) | |
| 2 | dfringc2.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
| 3 | dfringc2.b | . . 3 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) | |
| 4 | dfringc2.h | . . 3 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) | |
| 5 | 1, 2, 3, 4 | ringcval 20647 | . 2 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻)) |
| 6 | eqid 2737 | . . 3 ⊢ ((ExtStrCat‘𝑈) ↾cat 𝐻) = ((ExtStrCat‘𝑈) ↾cat 𝐻) | |
| 7 | fvexd 6921 | . . 3 ⊢ (𝜑 → (ExtStrCat‘𝑈) ∈ V) | |
| 8 | inex1g 5319 | . . . . 5 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V) | |
| 9 | 2, 8 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑈 ∩ Ring) ∈ V) |
| 10 | 3, 9 | eqeltrd 2841 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
| 11 | 3, 4 | rhmresfn 20648 | . . 3 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) |
| 12 | 6, 7, 10, 11 | rescval2 17872 | . 2 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) = (((ExtStrCat‘𝑈) ↾s 𝐵) sSet 〈(Hom ‘ndx), 𝐻〉)) |
| 13 | eqid 2737 | . . . 4 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈) | |
| 14 | eqidd 2738 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))) | |
| 15 | dfringc2.o | . . . . 5 ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈))) | |
| 16 | eqid 2737 | . . . . . 6 ⊢ (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈)) | |
| 17 | 13, 2, 16 | estrccofval 18173 | . . . . 5 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑m (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))) |
| 18 | 15, 17 | eqtrd 2777 | . . . 4 ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑m (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))) |
| 19 | 13, 2, 14, 18 | estrcval 18168 | . . 3 ⊢ (𝜑 → (ExtStrCat‘𝑈) = {〈(Base‘ndx), 𝑈〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))〉, 〈(comp‘ndx), · 〉}) |
| 20 | mpoexga 8102 | . . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑈 ∈ 𝑉) → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) ∈ V) | |
| 21 | 2, 2, 20 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) ∈ V) |
| 22 | fvexd 6921 | . . . 4 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V) | |
| 23 | 15, 22 | eqeltrd 2841 | . . 3 ⊢ (𝜑 → · ∈ V) |
| 24 | rhmfn 20499 | . . . . . 6 ⊢ RingHom Fn (Ring × Ring) | |
| 25 | fnfun 6668 | . . . . . 6 ⊢ ( RingHom Fn (Ring × Ring) → Fun RingHom ) | |
| 26 | 24, 25 | mp1i 13 | . . . . 5 ⊢ (𝜑 → Fun RingHom ) |
| 27 | sqxpexg 7775 | . . . . . 6 ⊢ (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V) | |
| 28 | 10, 27 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐵 × 𝐵) ∈ V) |
| 29 | resfunexg 7235 | . . . . 5 ⊢ ((Fun RingHom ∧ (𝐵 × 𝐵) ∈ V) → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V) | |
| 30 | 26, 28, 29 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V) |
| 31 | 4, 30 | eqeltrd 2841 | . . 3 ⊢ (𝜑 → 𝐻 ∈ V) |
| 32 | inss1 4237 | . . . 4 ⊢ (𝑈 ∩ Ring) ⊆ 𝑈 | |
| 33 | 3, 32 | eqsstrdi 4028 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝑈) |
| 34 | 19, 2, 21, 23, 31, 33 | estrres 18184 | . 2 ⊢ (𝜑 → (((ExtStrCat‘𝑈) ↾s 𝐵) sSet 〈(Hom ‘ndx), 𝐻〉) = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) |
| 35 | 5, 12, 34 | 3eqtrd 2781 | 1 ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 {ctp 4630 〈cop 4632 × cxp 5683 ↾ cres 5687 ∘ ccom 5689 Fun wfun 6555 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 1st c1st 8012 2nd c2nd 8013 ↑m cmap 8866 sSet csts 17200 ndxcnx 17230 Basecbs 17247 ↾s cress 17274 Hom chom 17308 compcco 17309 ↾cat cresc 17852 ExtStrCatcestrc 18166 Ringcrg 20230 RingHom crh 20469 RingCatcringc 20645 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-hom 17321 df-cco 17322 df-0g 17486 df-resc 17855 df-estrc 18167 df-mhm 18796 df-ghm 19231 df-mgp 20138 df-ur 20179 df-ring 20232 df-rhm 20472 df-ringc 20646 |
| This theorem is referenced by: rngcresringcat 20669 |
| Copyright terms: Public domain | W3C validator |