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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfrngc2 | Structured version Visualization version GIF version |
Description: Alternate definition of the category of non-unital rings (in a universe). (Contributed by AV, 16-Mar-2020.) |
Ref | Expression |
---|---|
dfrngc2.c | ⊢ 𝐶 = (RngCat‘𝑈) |
dfrngc2.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
dfrngc2.b | ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) |
dfrngc2.h | ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) |
dfrngc2.o | ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈))) |
Ref | Expression |
---|---|
dfrngc2 | ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrngc2.c | . . 3 ⊢ 𝐶 = (RngCat‘𝑈) | |
2 | dfrngc2.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
3 | dfrngc2.b | . . 3 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) | |
4 | dfrngc2.h | . . 3 ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) | |
5 | 1, 2, 3, 4 | rngcval 42287 | . 2 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻)) |
6 | eqid 2651 | . . . 4 ⊢ ((ExtStrCat‘𝑈) ↾cat 𝐻) = ((ExtStrCat‘𝑈) ↾cat 𝐻) | |
7 | fvexd 6241 | . . . 4 ⊢ (𝜑 → (ExtStrCat‘𝑈) ∈ V) | |
8 | inex1g 4834 | . . . . . 6 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V) | |
9 | 2, 8 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑈 ∩ Rng) ∈ V) |
10 | 3, 9 | eqeltrd 2730 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) |
11 | 3, 4 | rnghmresfn 42288 | . . . 4 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) |
12 | 6, 7, 10, 11 | rescval2 16535 | . . 3 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) = (((ExtStrCat‘𝑈) ↾s 𝐵) sSet 〈(Hom ‘ndx), 𝐻〉)) |
13 | eqid 2651 | . . . . 5 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈) | |
14 | eqidd 2652 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) | |
15 | dfrngc2.o | . . . . . 6 ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈))) | |
16 | eqid 2651 | . . . . . . 7 ⊢ (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈)) | |
17 | 13, 2, 16 | estrccofval 16816 | . . . . . 6 ⊢ (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 2685 | . . . . 5 ⊢ (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 16811 | . . . 4 ⊢ (𝜑 → (ExtStrCat‘𝑈) = {〈(Base‘ndx), 𝑈〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))〉, 〈(comp‘ndx), · 〉}) |
20 | 2, 2 | jca 553 | . . . . 5 ⊢ (𝜑 → (𝑈 ∈ 𝑉 ∧ 𝑈 ∈ 𝑉)) |
21 | eqid 2651 | . . . . . 6 ⊢ (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) | |
22 | 21 | mpt2exg 7290 | . . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑈 ∈ 𝑉) → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∈ V) |
23 | 20, 22 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∈ V) |
24 | fvexd 6241 | . . . . 5 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V) | |
25 | 15, 24 | eqeltrd 2730 | . . . 4 ⊢ (𝜑 → · ∈ V) |
26 | rnghmfn 42215 | . . . . . . 7 ⊢ RngHomo Fn (Rng × Rng) | |
27 | fnfun 6026 | . . . . . . 7 ⊢ ( RngHomo Fn (Rng × Rng) → Fun RngHomo ) | |
28 | 26, 27 | mp1i 13 | . . . . . 6 ⊢ (𝜑 → Fun RngHomo ) |
29 | sqxpexg 7005 | . . . . . . 7 ⊢ (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V) | |
30 | 10, 29 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐵 × 𝐵) ∈ V) |
31 | resfunexg 6520 | . . . . . 6 ⊢ ((Fun RngHomo ∧ (𝐵 × 𝐵) ∈ V) → ( RngHomo ↾ (𝐵 × 𝐵)) ∈ V) | |
32 | 28, 30, 31 | syl2anc 694 | . . . . 5 ⊢ (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) ∈ V) |
33 | 4, 32 | eqeltrd 2730 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ V) |
34 | inss1 3866 | . . . . . 6 ⊢ (𝑈 ∩ Rng) ⊆ 𝑈 | |
35 | 34 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑈 ∩ Rng) ⊆ 𝑈) |
36 | 3 | sseq1d 3665 | . . . . 5 ⊢ (𝜑 → (𝐵 ⊆ 𝑈 ↔ (𝑈 ∩ Rng) ⊆ 𝑈)) |
37 | 35, 36 | mpbird 247 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝑈) |
38 | 19, 2, 23, 25, 10, 33, 37 | estrres 16826 | . . 3 ⊢ (𝜑 → (((ExtStrCat‘𝑈) ↾s 𝐵) sSet 〈(Hom ‘ndx), 𝐻〉) = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) |
39 | 12, 38 | eqtrd 2685 | . 2 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) |
40 | 5, 39 | eqtrd 2685 | 1 ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ∩ cin 3606 ⊆ wss 3607 {ctp 4214 〈cop 4216 × cxp 5141 ↾ cres 5145 ∘ ccom 5147 Fun wfun 5920 Fn wfn 5921 ‘cfv 5926 (class class class)co 6690 ↦ cmpt2 6692 1st c1st 7208 2nd c2nd 7209 ↑𝑚 cmap 7899 ndxcnx 15901 sSet csts 15902 Basecbs 15904 ↾s cress 15905 Hom chom 15999 compcco 16000 ↾cat cresc 16515 ExtStrCatcestrc 16809 Rngcrng 42199 RngHomo crngh 42210 RngCatcrngc 42282 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-hom 16013 df-cco 16014 df-resc 16518 df-estrc 16810 df-rnghomo 42212 df-rngc 42284 |
This theorem is referenced by: rngcresringcat 42355 |
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