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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ringcid | Structured version Visualization version GIF version |
Description: The identity arrow in the category of unital rings is the identity function. (Contributed by AV, 14-Feb-2020.) (Revised by AV, 10-Mar-2020.) |
Ref | Expression |
---|---|
ringccat.c | ⊢ 𝐶 = (RingCat‘𝑈) |
ringcid.b | ⊢ 𝐵 = (Base‘𝐶) |
ringcid.o | ⊢ 1 = (Id‘𝐶) |
ringcid.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
ringcid.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ringcid.s | ⊢ 𝑆 = (Base‘𝑋) |
Ref | Expression |
---|---|
ringcid | ⊢ (𝜑 → ( 1 ‘𝑋) = ( I ↾ 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringcid.o | . . . 4 ⊢ 1 = (Id‘𝐶) | |
2 | ringccat.c | . . . . . 6 ⊢ 𝐶 = (RingCat‘𝑈) | |
3 | ringcid.u | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
4 | eqidd 2761 | . . . . . 6 ⊢ (𝜑 → (𝑈 ∩ Ring) = (𝑈 ∩ Ring)) | |
5 | eqidd 2761 | . . . . . 6 ⊢ (𝜑 → ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring))) = ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))) | |
6 | 2, 3, 4, 5 | ringcval 42536 | . . . . 5 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring))))) |
7 | 6 | fveq2d 6357 | . . . 4 ⊢ (𝜑 → (Id‘𝐶) = (Id‘((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))))) |
8 | 1, 7 | syl5eq 2806 | . . 3 ⊢ (𝜑 → 1 = (Id‘((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))))) |
9 | 8 | fveq1d 6355 | . 2 ⊢ (𝜑 → ( 1 ‘𝑋) = ((Id‘((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))))‘𝑋)) |
10 | eqid 2760 | . . 3 ⊢ ((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))) = ((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))) | |
11 | eqid 2760 | . . . 4 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈) | |
12 | incom 3948 | . . . . 5 ⊢ (𝑈 ∩ Ring) = (Ring ∩ 𝑈) | |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑈 ∩ Ring) = (Ring ∩ 𝑈)) |
14 | 11, 3, 13, 5 | rhmsubcsetc 42551 | . . 3 ⊢ (𝜑 → ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring))) ∈ (Subcat‘(ExtStrCat‘𝑈))) |
15 | 4, 5 | rhmresfn 42537 | . . 3 ⊢ (𝜑 → ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring))) Fn ((𝑈 ∩ Ring) × (𝑈 ∩ Ring))) |
16 | eqid 2760 | . . 3 ⊢ (Id‘(ExtStrCat‘𝑈)) = (Id‘(ExtStrCat‘𝑈)) | |
17 | ringcid.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
18 | ringcid.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
19 | 2, 18, 3 | ringcbas 42539 | . . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) |
20 | 19 | eleq2d 2825 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ (𝑈 ∩ Ring))) |
21 | 17, 20 | mpbid 222 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑈 ∩ Ring)) |
22 | 10, 14, 15, 16, 21 | subcid 16728 | . 2 ⊢ (𝜑 → ((Id‘(ExtStrCat‘𝑈))‘𝑋) = ((Id‘((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))))‘𝑋)) |
23 | elinel1 3942 | . . . . . 6 ⊢ (𝑋 ∈ (𝑈 ∩ Ring) → 𝑋 ∈ 𝑈) | |
24 | 20, 23 | syl6bi 243 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝐵 → 𝑋 ∈ 𝑈)) |
25 | 17, 24 | mpd 15 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑈) |
26 | 11, 16, 3, 25 | estrcid 16995 | . . 3 ⊢ (𝜑 → ((Id‘(ExtStrCat‘𝑈))‘𝑋) = ( I ↾ (Base‘𝑋))) |
27 | ringcid.s | . . . . . 6 ⊢ 𝑆 = (Base‘𝑋) | |
28 | 27 | eqcomi 2769 | . . . . 5 ⊢ (Base‘𝑋) = 𝑆 |
29 | 28 | a1i 11 | . . . 4 ⊢ (𝜑 → (Base‘𝑋) = 𝑆) |
30 | 29 | reseq2d 5551 | . . 3 ⊢ (𝜑 → ( I ↾ (Base‘𝑋)) = ( I ↾ 𝑆)) |
31 | 26, 30 | eqtrd 2794 | . 2 ⊢ (𝜑 → ((Id‘(ExtStrCat‘𝑈))‘𝑋) = ( I ↾ 𝑆)) |
32 | 9, 22, 31 | 3eqtr2d 2800 | 1 ⊢ (𝜑 → ( 1 ‘𝑋) = ( I ↾ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1632 ∈ wcel 2139 ∩ cin 3714 I cid 5173 × cxp 5264 ↾ cres 5268 ‘cfv 6049 (class class class)co 6814 Basecbs 16079 Idccid 16547 ↾cat cresc 16689 ExtStrCatcestrc 16983 Ringcrg 18767 RingHom crh 18934 RingCatcringc 42531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-1st 7334 df-2nd 7335 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-oadd 7734 df-er 7913 df-map 8027 df-pm 8028 df-ixp 8077 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-nn 11233 df-2 11291 df-3 11292 df-4 11293 df-5 11294 df-6 11295 df-7 11296 df-8 11297 df-9 11298 df-n0 11505 df-z 11590 df-dec 11706 df-uz 11900 df-fz 12540 df-struct 16081 df-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-ress 16087 df-plusg 16176 df-hom 16188 df-cco 16189 df-0g 16324 df-cat 16550 df-cid 16551 df-homf 16552 df-ssc 16691 df-resc 16692 df-subc 16693 df-estrc 16984 df-mgm 17463 df-sgrp 17505 df-mnd 17516 df-mhm 17556 df-grp 17646 df-ghm 17879 df-mgp 18710 df-ur 18722 df-ring 18769 df-rnghom 18937 df-ringc 42533 |
This theorem is referenced by: ringcsect 42559 funcringcsetcALTV2lem7 42570 srhmsubc 42604 |
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