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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ringchomfval | Structured version Visualization version GIF version |
Description: Set of arrows of the category of unital rings (in a universe). (Contributed by AV, 14-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
Ref | Expression |
---|---|
ringcbas.c | ⊢ 𝐶 = (RingCat‘𝑈) |
ringcbas.b | ⊢ 𝐵 = (Base‘𝐶) |
ringcbas.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
ringchomfval.h | ⊢ 𝐻 = (Hom ‘𝐶) |
Ref | Expression |
---|---|
ringchomfval | ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringchomfval.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
2 | ringcbas.c | . . . . 5 ⊢ 𝐶 = (RingCat‘𝑈) | |
3 | ringcbas.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
4 | ringcbas.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
5 | 2, 4, 3 | ringcbas 43646 | . . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) |
6 | eqidd 2773 | . . . . 5 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) = ( RingHom ↾ (𝐵 × 𝐵))) | |
7 | 2, 3, 5, 6 | ringcval 43643 | . . . 4 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ (𝐵 × 𝐵)))) |
8 | 7 | fveq2d 6497 | . . 3 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ (𝐵 × 𝐵))))) |
9 | 1, 8 | syl5eq 2820 | . 2 ⊢ (𝜑 → 𝐻 = (Hom ‘((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ (𝐵 × 𝐵))))) |
10 | eqid 2772 | . . 3 ⊢ ((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ (𝐵 × 𝐵))) = ((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ (𝐵 × 𝐵))) | |
11 | eqid 2772 | . . 3 ⊢ (Base‘(ExtStrCat‘𝑈)) = (Base‘(ExtStrCat‘𝑈)) | |
12 | fvexd 6508 | . . 3 ⊢ (𝜑 → (ExtStrCat‘𝑈) ∈ V) | |
13 | 5, 6 | rhmresfn 43644 | . . 3 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵)) |
14 | inss1 4086 | . . . . 5 ⊢ (𝑈 ∩ Ring) ⊆ 𝑈 | |
15 | 14 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑈 ∩ Ring) ⊆ 𝑈) |
16 | eqid 2772 | . . . . . 6 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈) | |
17 | 16, 3 | estrcbas 17227 | . . . . 5 ⊢ (𝜑 → 𝑈 = (Base‘(ExtStrCat‘𝑈))) |
18 | 17 | eqcomd 2778 | . . . 4 ⊢ (𝜑 → (Base‘(ExtStrCat‘𝑈)) = 𝑈) |
19 | 15, 5, 18 | 3sstr4d 3898 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ (Base‘(ExtStrCat‘𝑈))) |
20 | 10, 11, 12, 13, 19 | reschom 16952 | . 2 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) = (Hom ‘((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ (𝐵 × 𝐵))))) |
21 | 9, 20 | eqtr4d 2811 | 1 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 Vcvv 3409 ∩ cin 3822 ⊆ wss 3823 × cxp 5399 ↾ cres 5403 ‘cfv 6182 (class class class)co 6970 Basecbs 16333 Hom chom 16426 ↾cat cresc 16930 ExtStrCatcestrc 17224 Ringcrg 19014 RingHom crh 19181 RingCatcringc 43638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7495 df-2nd 7496 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-1o 7899 df-oadd 7903 df-er 8083 df-map 8202 df-en 8301 df-dom 8302 df-sdom 8303 df-fin 8304 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-nn 11434 df-2 11497 df-3 11498 df-4 11499 df-5 11500 df-6 11501 df-7 11502 df-8 11503 df-9 11504 df-n0 11702 df-z 11788 df-dec 11906 df-uz 12053 df-fz 12703 df-struct 16335 df-ndx 16336 df-slot 16337 df-base 16339 df-sets 16340 df-ress 16341 df-plusg 16428 df-hom 16439 df-cco 16440 df-0g 16565 df-resc 16933 df-estrc 17225 df-mhm 17797 df-ghm 18121 df-mgp 18957 df-ur 18969 df-ring 19016 df-rnghom 19184 df-ringc 43640 |
This theorem is referenced by: ringchom 43648 ringchomfeqhom 43650 ringccofval 43651 rhmsubcsetclem1 43656 rhmsubcrngclem1 43662 funcringcsetc 43670 irinitoringc 43704 |
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