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 2821 | . . . . . 6 ⊢ (𝜑 → (𝑈 ∩ Ring) = (𝑈 ∩ Ring)) | |
5 | eqidd 2821 | . . . . . 6 ⊢ (𝜑 → ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring))) = ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))) | |
6 | 2, 3, 4, 5 | ringcval 44359 | . . . . 5 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring))))) |
7 | 6 | fveq2d 6655 | . . . 4 ⊢ (𝜑 → (Id‘𝐶) = (Id‘((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))))) |
8 | 1, 7 | syl5eq 2867 | . . 3 ⊢ (𝜑 → 1 = (Id‘((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))))) |
9 | 8 | fveq1d 6653 | . 2 ⊢ (𝜑 → ( 1 ‘𝑋) = ((Id‘((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))))‘𝑋)) |
10 | eqid 2820 | . . 3 ⊢ ((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))) = ((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))) | |
11 | eqid 2820 | . . . 4 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈) | |
12 | incom 4161 | . . . . 5 ⊢ (𝑈 ∩ Ring) = (Ring ∩ 𝑈) | |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑈 ∩ Ring) = (Ring ∩ 𝑈)) |
14 | 11, 3, 13, 5 | rhmsubcsetc 44374 | . . 3 ⊢ (𝜑 → ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring))) ∈ (Subcat‘(ExtStrCat‘𝑈))) |
15 | 4, 5 | rhmresfn 44360 | . . 3 ⊢ (𝜑 → ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring))) Fn ((𝑈 ∩ Ring) × (𝑈 ∩ Ring))) |
16 | eqid 2820 | . . 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 44362 | . . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) |
20 | 19 | eleq2d 2896 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ (𝑈 ∩ Ring))) |
21 | 17, 20 | mpbid 234 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑈 ∩ Ring)) |
22 | 10, 14, 15, 16, 21 | subcid 17095 | . 2 ⊢ (𝜑 → ((Id‘(ExtStrCat‘𝑈))‘𝑋) = ((Id‘((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))))‘𝑋)) |
23 | elinel1 4155 | . . . . . 6 ⊢ (𝑋 ∈ (𝑈 ∩ Ring) → 𝑋 ∈ 𝑈) | |
24 | 20, 23 | syl6bi 255 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝐵 → 𝑋 ∈ 𝑈)) |
25 | 17, 24 | mpd 15 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑈) |
26 | 11, 16, 3, 25 | estrcid 17362 | . . 3 ⊢ (𝜑 → ((Id‘(ExtStrCat‘𝑈))‘𝑋) = ( I ↾ (Base‘𝑋))) |
27 | ringcid.s | . . . . . 6 ⊢ 𝑆 = (Base‘𝑋) | |
28 | 27 | eqcomi 2829 | . . . . 5 ⊢ (Base‘𝑋) = 𝑆 |
29 | 28 | a1i 11 | . . . 4 ⊢ (𝜑 → (Base‘𝑋) = 𝑆) |
30 | 29 | reseq2d 5834 | . . 3 ⊢ (𝜑 → ( I ↾ (Base‘𝑋)) = ( I ↾ 𝑆)) |
31 | 26, 30 | eqtrd 2855 | . 2 ⊢ (𝜑 → ((Id‘(ExtStrCat‘𝑈))‘𝑋) = ( I ↾ 𝑆)) |
32 | 9, 22, 31 | 3eqtr2d 2861 | 1 ⊢ (𝜑 → ( 1 ‘𝑋) = ( I ↾ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∩ cin 3918 I cid 5440 × cxp 5534 ↾ cres 5538 ‘cfv 6336 (class class class)co 7137 Basecbs 16461 Idccid 16914 ↾cat cresc 17056 ExtStrCatcestrc 17350 Ringcrg 19275 RingHom crh 19442 RingCatcringc 44354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-rep 5171 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-cnex 10574 ax-resscn 10575 ax-1cn 10576 ax-icn 10577 ax-addcl 10578 ax-addrcl 10579 ax-mulcl 10580 ax-mulrcl 10581 ax-mulcom 10582 ax-addass 10583 ax-mulass 10584 ax-distr 10585 ax-i2m1 10586 ax-1ne0 10587 ax-1rid 10588 ax-rnegex 10589 ax-rrecex 10590 ax-cnre 10591 ax-pre-lttri 10592 ax-pre-lttrn 10593 ax-pre-ltadd 10594 ax-pre-mulgt0 10595 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-nel 3119 df-ral 3138 df-rex 3139 df-reu 3140 df-rmo 3141 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-int 4858 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-we 5497 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7095 df-ov 7140 df-oprab 7141 df-mpo 7142 df-om 7562 df-1st 7670 df-2nd 7671 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-1o 8083 df-oadd 8087 df-er 8270 df-map 8389 df-pm 8390 df-ixp 8443 df-en 8491 df-dom 8492 df-sdom 8493 df-fin 8494 df-pnf 10658 df-mnf 10659 df-xr 10660 df-ltxr 10661 df-le 10662 df-sub 10853 df-neg 10854 df-nn 11620 df-2 11682 df-3 11683 df-4 11684 df-5 11685 df-6 11686 df-7 11687 df-8 11688 df-9 11689 df-n0 11880 df-z 11964 df-dec 12081 df-uz 12226 df-fz 12878 df-struct 16463 df-ndx 16464 df-slot 16465 df-base 16467 df-sets 16468 df-ress 16469 df-plusg 16556 df-hom 16567 df-cco 16568 df-0g 16693 df-cat 16917 df-cid 16918 df-homf 16919 df-ssc 17058 df-resc 17059 df-subc 17060 df-estrc 17351 df-mgm 17830 df-sgrp 17879 df-mnd 17890 df-mhm 17934 df-grp 18084 df-ghm 18334 df-mgp 19218 df-ur 19230 df-ring 19277 df-rnghom 19445 df-ringc 44356 |
This theorem is referenced by: ringcsect 44382 funcringcsetcALTV2lem7 44393 srhmsubc 44427 |
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