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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 20664 | . 2 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻)) |
6 | eqid 2735 | . . 3 ⊢ ((ExtStrCat‘𝑈) ↾cat 𝐻) = ((ExtStrCat‘𝑈) ↾cat 𝐻) | |
7 | fvexd 6922 | . . 3 ⊢ (𝜑 → (ExtStrCat‘𝑈) ∈ V) | |
8 | inex1g 5325 | . . . . 5 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V) | |
9 | 2, 8 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑈 ∩ Ring) ∈ V) |
10 | 3, 9 | eqeltrd 2839 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
11 | 3, 4 | rhmresfn 20665 | . . 3 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) |
12 | 6, 7, 10, 11 | rescval2 17876 | . 2 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) = (((ExtStrCat‘𝑈) ↾s 𝐵) sSet 〈(Hom ‘ndx), 𝐻〉)) |
13 | eqid 2735 | . . . 4 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈) | |
14 | eqidd 2736 | . . . 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 2735 | . . . . . 6 ⊢ (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈)) | |
17 | 13, 2, 16 | estrccofval 18184 | . . . . 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 2775 | . . . 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 18179 | . . 3 ⊢ (𝜑 → (ExtStrCat‘𝑈) = {〈(Base‘ndx), 𝑈〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))〉, 〈(comp‘ndx), · 〉}) |
20 | mpoexga 8101 | . . . 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 6922 | . . . 4 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V) | |
23 | 15, 22 | eqeltrd 2839 | . . 3 ⊢ (𝜑 → · ∈ V) |
24 | rhmfn 20516 | . . . . . 6 ⊢ RingHom Fn (Ring × Ring) | |
25 | fnfun 6669 | . . . . . 6 ⊢ ( RingHom Fn (Ring × Ring) → Fun RingHom ) | |
26 | 24, 25 | mp1i 13 | . . . . 5 ⊢ (𝜑 → Fun RingHom ) |
27 | sqxpexg 7774 | . . . . . 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 2839 | . . 3 ⊢ (𝜑 → 𝐻 ∈ V) |
32 | inss1 4245 | . . . 4 ⊢ (𝑈 ∩ Ring) ⊆ 𝑈 | |
33 | 3, 32 | eqsstrdi 4050 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝑈) |
34 | 19, 2, 21, 23, 31, 33 | estrres 18195 | . 2 ⊢ (𝜑 → (((ExtStrCat‘𝑈) ↾s 𝐵) sSet 〈(Hom ‘ndx), 𝐻〉) = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) |
35 | 5, 12, 34 | 3eqtrd 2779 | 1 ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∩ cin 3962 {ctp 4635 〈cop 4637 × cxp 5687 ↾ cres 5691 ∘ ccom 5693 Fun wfun 6557 Fn wfn 6558 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 1st c1st 8011 2nd c2nd 8012 ↑m cmap 8865 sSet csts 17197 ndxcnx 17227 Basecbs 17245 ↾s cress 17274 Hom chom 17309 compcco 17310 ↾cat cresc 17856 ExtStrCatcestrc 18177 Ringcrg 20251 RingHom crh 20486 RingCatcringc 20662 |
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-tp 4636 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-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 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-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-hom 17322 df-cco 17323 df-0g 17488 df-resc 17859 df-estrc 18178 df-mhm 18809 df-ghm 19244 df-mgp 20153 df-ur 20200 df-ring 20253 df-rhm 20489 df-ringc 20663 |
This theorem is referenced by: rngcresringcat 20686 |
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