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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngchomfval | Structured version Visualization version GIF version |
Description: Set of arrows of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
Ref | Expression |
---|---|
rngcbas.c | ⊢ 𝐶 = (RngCat‘𝑈) |
rngcbas.b | ⊢ 𝐵 = (Base‘𝐶) |
rngcbas.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
rngchomfval.h | ⊢ 𝐻 = (Hom ‘𝐶) |
Ref | Expression |
---|---|
rngchomfval | ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngchomfval.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
2 | rngcbas.c | . . . . 5 ⊢ 𝐶 = (RngCat‘𝑈) | |
3 | rngcbas.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
4 | rngcbas.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
5 | 2, 4, 3 | rngcbas 46765 | . . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) |
6 | eqidd 2734 | . . . . 5 ⊢ (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) = ( RngHomo ↾ (𝐵 × 𝐵))) | |
7 | 2, 3, 5, 6 | rngcval 46762 | . . . 4 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat ( RngHomo ↾ (𝐵 × 𝐵)))) |
8 | 7 | fveq2d 6892 | . . 3 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘((ExtStrCat‘𝑈) ↾cat ( RngHomo ↾ (𝐵 × 𝐵))))) |
9 | 1, 8 | eqtrid 2785 | . 2 ⊢ (𝜑 → 𝐻 = (Hom ‘((ExtStrCat‘𝑈) ↾cat ( RngHomo ↾ (𝐵 × 𝐵))))) |
10 | eqid 2733 | . . 3 ⊢ ((ExtStrCat‘𝑈) ↾cat ( RngHomo ↾ (𝐵 × 𝐵))) = ((ExtStrCat‘𝑈) ↾cat ( RngHomo ↾ (𝐵 × 𝐵))) | |
11 | eqid 2733 | . . 3 ⊢ (Base‘(ExtStrCat‘𝑈)) = (Base‘(ExtStrCat‘𝑈)) | |
12 | fvexd 6903 | . . 3 ⊢ (𝜑 → (ExtStrCat‘𝑈) ∈ V) | |
13 | 5, 6 | rnghmresfn 46763 | . . 3 ⊢ (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵)) |
14 | inss1 4227 | . . . . 5 ⊢ (𝑈 ∩ Rng) ⊆ 𝑈 | |
15 | 14 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑈 ∩ Rng) ⊆ 𝑈) |
16 | eqid 2733 | . . . . . 6 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈) | |
17 | 16, 3 | estrcbas 18072 | . . . . 5 ⊢ (𝜑 → 𝑈 = (Base‘(ExtStrCat‘𝑈))) |
18 | 17 | eqcomd 2739 | . . . 4 ⊢ (𝜑 → (Base‘(ExtStrCat‘𝑈)) = 𝑈) |
19 | 15, 5, 18 | 3sstr4d 4028 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ (Base‘(ExtStrCat‘𝑈))) |
20 | 10, 11, 12, 13, 19 | reschom 17774 | . 2 ⊢ (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) = (Hom ‘((ExtStrCat‘𝑈) ↾cat ( RngHomo ↾ (𝐵 × 𝐵))))) |
21 | 9, 20 | eqtr4d 2776 | 1 ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∩ cin 3946 ⊆ wss 3947 × cxp 5673 ↾ cres 5677 ‘cfv 6540 (class class class)co 7404 Basecbs 17140 Hom chom 17204 ↾cat cresc 17751 ExtStrCatcestrc 18069 Rngcrng 46583 RngHomo crngh 46617 RngCatcrngc 46757 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-hom 17217 df-cco 17218 df-resc 17754 df-estrc 18070 df-rnghomo 46619 df-rngc 46759 |
This theorem is referenced by: rngchom 46767 rngchomfeqhom 46769 rngccofval 46770 rnghmsubcsetclem1 46775 rngcifuestrc 46797 funcrngcsetc 46798 rhmsubcrngc 46829 rhmsubc 46890 |
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