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 44210 | . . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) |
6 | eqidd 2819 | . . . . 5 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) = ( RingHom ↾ (𝐵 × 𝐵))) | |
7 | 2, 3, 5, 6 | ringcval 44207 | . . . 4 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ (𝐵 × 𝐵)))) |
8 | 7 | fveq2d 6667 | . . 3 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ (𝐵 × 𝐵))))) |
9 | 1, 8 | syl5eq 2865 | . 2 ⊢ (𝜑 → 𝐻 = (Hom ‘((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ (𝐵 × 𝐵))))) |
10 | eqid 2818 | . . 3 ⊢ ((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ (𝐵 × 𝐵))) = ((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ (𝐵 × 𝐵))) | |
11 | eqid 2818 | . . 3 ⊢ (Base‘(ExtStrCat‘𝑈)) = (Base‘(ExtStrCat‘𝑈)) | |
12 | fvexd 6678 | . . 3 ⊢ (𝜑 → (ExtStrCat‘𝑈) ∈ V) | |
13 | 5, 6 | rhmresfn 44208 | . . 3 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵)) |
14 | inss1 4202 | . . . . 5 ⊢ (𝑈 ∩ Ring) ⊆ 𝑈 | |
15 | 14 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑈 ∩ Ring) ⊆ 𝑈) |
16 | eqid 2818 | . . . . . 6 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈) | |
17 | 16, 3 | estrcbas 17363 | . . . . 5 ⊢ (𝜑 → 𝑈 = (Base‘(ExtStrCat‘𝑈))) |
18 | 17 | eqcomd 2824 | . . . 4 ⊢ (𝜑 → (Base‘(ExtStrCat‘𝑈)) = 𝑈) |
19 | 15, 5, 18 | 3sstr4d 4011 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ (Base‘(ExtStrCat‘𝑈))) |
20 | 10, 11, 12, 13, 19 | reschom 17088 | . 2 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) = (Hom ‘((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ (𝐵 × 𝐵))))) |
21 | 9, 20 | eqtr4d 2856 | 1 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∩ cin 3932 ⊆ wss 3933 × cxp 5546 ↾ cres 5550 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 Hom chom 16564 ↾cat cresc 17066 ExtStrCatcestrc 17360 Ringcrg 19226 RingHom crh 19393 RingCatcringc 44202 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 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 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-hom 16577 df-cco 16578 df-0g 16703 df-resc 17069 df-estrc 17361 df-mhm 17944 df-ghm 18294 df-mgp 19169 df-ur 19181 df-ring 19228 df-rnghom 19396 df-ringc 44204 |
This theorem is referenced by: ringchom 44212 ringchomfeqhom 44214 ringccofval 44215 rhmsubcsetclem1 44220 rhmsubcrngclem1 44226 funcringcsetc 44234 irinitoringc 44268 |
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