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Mathbox for Alexander van der Vekens |
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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 44586 | . 2 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻)) |
6 | eqid 2798 | . . 3 ⊢ ((ExtStrCat‘𝑈) ↾cat 𝐻) = ((ExtStrCat‘𝑈) ↾cat 𝐻) | |
7 | fvexd 6660 | . . 3 ⊢ (𝜑 → (ExtStrCat‘𝑈) ∈ V) | |
8 | inex1g 5187 | . . . . 5 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V) | |
9 | 2, 8 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑈 ∩ Rng) ∈ V) |
10 | 3, 9 | eqeltrd 2890 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
11 | 3, 4 | rnghmresfn 44587 | . . 3 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) |
12 | 6, 7, 10, 11 | rescval2 17090 | . 2 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) = (((ExtStrCat‘𝑈) ↾s 𝐵) sSet 〈(Hom ‘ndx), 𝐻〉)) |
13 | eqid 2798 | . . . 4 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈) | |
14 | eqidd 2799 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))) | |
15 | dfrngc2.o | . . . . 5 ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈))) | |
16 | eqid 2798 | . . . . . 6 ⊢ (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈)) | |
17 | 13, 2, 16 | estrccofval 17371 | . . . . 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 2833 | . . . 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 17366 | . . 3 ⊢ (𝜑 → (ExtStrCat‘𝑈) = {〈(Base‘ndx), 𝑈〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))〉, 〈(comp‘ndx), · 〉}) |
20 | mpoexga 7758 | . . . 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 6660 | . . . 4 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V) | |
23 | 15, 22 | eqeltrd 2890 | . . 3 ⊢ (𝜑 → · ∈ V) |
24 | rnghmfn 44514 | . . . . . 6 ⊢ RngHomo Fn (Rng × Rng) | |
25 | fnfun 6423 | . . . . . 6 ⊢ ( RngHomo Fn (Rng × Rng) → Fun RngHomo ) | |
26 | 24, 25 | mp1i 13 | . . . . 5 ⊢ (𝜑 → Fun RngHomo ) |
27 | sqxpexg 7457 | . . . . . 6 ⊢ (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V) | |
28 | 10, 27 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐵 × 𝐵) ∈ V) |
29 | resfunexg 6955 | . . . . 5 ⊢ ((Fun RngHomo ∧ (𝐵 × 𝐵) ∈ V) → ( RngHomo ↾ (𝐵 × 𝐵)) ∈ V) | |
30 | 26, 28, 29 | syl2anc 587 | . . . 4 ⊢ (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) ∈ V) |
31 | 4, 30 | eqeltrd 2890 | . . 3 ⊢ (𝜑 → 𝐻 ∈ V) |
32 | inss1 4155 | . . . 4 ⊢ (𝑈 ∩ Rng) ⊆ 𝑈 | |
33 | 3, 32 | eqsstrdi 3969 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝑈) |
34 | 19, 2, 21, 23, 31, 33 | estrres 17381 | . 2 ⊢ (𝜑 → (((ExtStrCat‘𝑈) ↾s 𝐵) sSet 〈(Hom ‘ndx), 𝐻〉) = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) |
35 | 5, 12, 34 | 3eqtrd 2837 | 1 ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∩ cin 3880 {ctp 4529 〈cop 4531 × cxp 5517 ↾ cres 5521 ∘ ccom 5523 Fun wfun 6318 Fn wfn 6319 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 1st c1st 7669 2nd c2nd 7670 ↑m cmap 8389 ndxcnx 16472 sSet csts 16473 Basecbs 16475 ↾s cress 16476 Hom chom 16568 compcco 16569 ↾cat cresc 17070 ExtStrCatcestrc 17364 Rngcrng 44498 RngHomo crngh 44509 RngCatcrngc 44581 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-hom 16581 df-cco 16582 df-resc 17073 df-estrc 17365 df-rnghomo 44511 df-rngc 44583 |
This theorem is referenced by: rngcresringcat 44654 |
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