Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 45182 | . 2 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻)) |
6 | eqid 2736 | . . 3 ⊢ ((ExtStrCat‘𝑈) ↾cat 𝐻) = ((ExtStrCat‘𝑈) ↾cat 𝐻) | |
7 | fvexd 6710 | . . 3 ⊢ (𝜑 → (ExtStrCat‘𝑈) ∈ V) | |
8 | inex1g 5197 | . . . . 5 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Ring) ∈ V) | |
9 | 2, 8 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑈 ∩ Ring) ∈ V) |
10 | 3, 9 | eqeltrd 2831 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
11 | 3, 4 | rhmresfn 45183 | . . 3 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) |
12 | 6, 7, 10, 11 | rescval2 17287 | . 2 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) = (((ExtStrCat‘𝑈) ↾s 𝐵) sSet 〈(Hom ‘ndx), 𝐻〉)) |
13 | eqid 2736 | . . . 4 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈) | |
14 | eqidd 2737 | . . . 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 2736 | . . . . . 6 ⊢ (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈)) | |
17 | 13, 2, 16 | estrccofval 17590 | . . . . 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 2771 | . . . 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 17585 | . . 3 ⊢ (𝜑 → (ExtStrCat‘𝑈) = {〈(Base‘ndx), 𝑈〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))〉, 〈(comp‘ndx), · 〉}) |
20 | mpoexga 7826 | . . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑈 ∈ 𝑉) → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) ∈ V) | |
21 | 2, 2, 20 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) ∈ V) |
22 | fvexd 6710 | . . . 4 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V) | |
23 | 15, 22 | eqeltrd 2831 | . . 3 ⊢ (𝜑 → · ∈ V) |
24 | rhmfn 45092 | . . . . . 6 ⊢ RingHom Fn (Ring × Ring) | |
25 | fnfun 6457 | . . . . . 6 ⊢ ( RingHom Fn (Ring × Ring) → Fun RingHom ) | |
26 | 24, 25 | mp1i 13 | . . . . 5 ⊢ (𝜑 → Fun RingHom ) |
27 | sqxpexg 7518 | . . . . . 6 ⊢ (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V) | |
28 | 10, 27 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐵 × 𝐵) ∈ V) |
29 | resfunexg 7009 | . . . . 5 ⊢ ((Fun RingHom ∧ (𝐵 × 𝐵) ∈ V) → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V) | |
30 | 26, 28, 29 | syl2anc 587 | . . . 4 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) ∈ V) |
31 | 4, 30 | eqeltrd 2831 | . . 3 ⊢ (𝜑 → 𝐻 ∈ V) |
32 | inss1 4129 | . . . 4 ⊢ (𝑈 ∩ Ring) ⊆ 𝑈 | |
33 | 3, 32 | eqsstrdi 3941 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝑈) |
34 | 19, 2, 21, 23, 31, 33 | estrres 17600 | . 2 ⊢ (𝜑 → (((ExtStrCat‘𝑈) ↾s 𝐵) sSet 〈(Hom ‘ndx), 𝐻〉) = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) |
35 | 5, 12, 34 | 3eqtrd 2775 | 1 ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 Vcvv 3398 ∩ cin 3852 {ctp 4531 〈cop 4533 × cxp 5534 ↾ cres 5538 ∘ ccom 5540 Fun wfun 6352 Fn wfn 6353 ‘cfv 6358 (class class class)co 7191 ∈ cmpo 7193 1st c1st 7737 2nd c2nd 7738 ↑m cmap 8486 ndxcnx 16663 sSet csts 16664 Basecbs 16666 ↾s cress 16667 Hom chom 16760 compcco 16761 ↾cat cresc 17267 ExtStrCatcestrc 17583 Ringcrg 19516 RingHom crh 19686 RingCatcringc 45177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-hom 16773 df-cco 16774 df-0g 16900 df-resc 17270 df-estrc 17584 df-mhm 18172 df-ghm 18574 df-mgp 19459 df-ur 19471 df-ring 19518 df-rnghom 19689 df-ringc 45179 |
This theorem is referenced by: rngcresringcat 45204 |
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