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 2737 | . . . . . 6 ⊢ (𝜑 → (𝑈 ∩ Ring) = (𝑈 ∩ Ring)) | |
5 | eqidd 2737 | . . . . . 6 ⊢ (𝜑 → ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring))) = ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))) | |
6 | 2, 3, 4, 5 | ringcval 45182 | . . . . 5 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring))))) |
7 | 6 | fveq2d 6699 | . . . 4 ⊢ (𝜑 → (Id‘𝐶) = (Id‘((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))))) |
8 | 1, 7 | syl5eq 2783 | . . 3 ⊢ (𝜑 → 1 = (Id‘((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))))) |
9 | 8 | fveq1d 6697 | . 2 ⊢ (𝜑 → ( 1 ‘𝑋) = ((Id‘((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))))‘𝑋)) |
10 | eqid 2736 | . . 3 ⊢ ((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))) = ((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))) | |
11 | eqid 2736 | . . . 4 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈) | |
12 | incom 4101 | . . . . 5 ⊢ (𝑈 ∩ Ring) = (Ring ∩ 𝑈) | |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑈 ∩ Ring) = (Ring ∩ 𝑈)) |
14 | 11, 3, 13, 5 | rhmsubcsetc 45197 | . . 3 ⊢ (𝜑 → ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring))) ∈ (Subcat‘(ExtStrCat‘𝑈))) |
15 | 4, 5 | rhmresfn 45183 | . . 3 ⊢ (𝜑 → ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring))) Fn ((𝑈 ∩ Ring) × (𝑈 ∩ Ring))) |
16 | eqid 2736 | . . 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 45185 | . . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) |
20 | 19 | eleq2d 2816 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ (𝑈 ∩ Ring))) |
21 | 17, 20 | mpbid 235 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑈 ∩ Ring)) |
22 | 10, 14, 15, 16, 21 | subcid 17307 | . 2 ⊢ (𝜑 → ((Id‘(ExtStrCat‘𝑈))‘𝑋) = ((Id‘((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))))‘𝑋)) |
23 | elinel1 4095 | . . . . . 6 ⊢ (𝑋 ∈ (𝑈 ∩ Ring) → 𝑋 ∈ 𝑈) | |
24 | 20, 23 | syl6bi 256 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝐵 → 𝑋 ∈ 𝑈)) |
25 | 17, 24 | mpd 15 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑈) |
26 | 11, 16, 3, 25 | estrcid 17595 | . . 3 ⊢ (𝜑 → ((Id‘(ExtStrCat‘𝑈))‘𝑋) = ( I ↾ (Base‘𝑋))) |
27 | ringcid.s | . . . . . 6 ⊢ 𝑆 = (Base‘𝑋) | |
28 | 27 | eqcomi 2745 | . . . . 5 ⊢ (Base‘𝑋) = 𝑆 |
29 | 28 | a1i 11 | . . . 4 ⊢ (𝜑 → (Base‘𝑋) = 𝑆) |
30 | 29 | reseq2d 5836 | . . 3 ⊢ (𝜑 → ( I ↾ (Base‘𝑋)) = ( I ↾ 𝑆)) |
31 | 26, 30 | eqtrd 2771 | . 2 ⊢ (𝜑 → ((Id‘(ExtStrCat‘𝑈))‘𝑋) = ( I ↾ 𝑆)) |
32 | 9, 22, 31 | 3eqtr2d 2777 | 1 ⊢ (𝜑 → ( 1 ‘𝑋) = ( I ↾ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ∩ cin 3852 I cid 5439 × cxp 5534 ↾ cres 5538 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 Idccid 17122 ↾cat cresc 17267 ExtStrCatcestrc 17583 Ringcrg 19516 RingHom crh 19686 RingCatcringc 45177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 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 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-pm 8489 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-hom 16773 df-cco 16774 df-0g 16900 df-cat 17125 df-cid 17126 df-homf 17127 df-ssc 17269 df-resc 17270 df-subc 17271 df-estrc 17584 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-grp 18322 df-ghm 18574 df-mgp 19459 df-ur 19471 df-ring 19518 df-rnghom 19689 df-ringc 45179 |
This theorem is referenced by: ringcsect 45205 funcringcsetcALTV2lem7 45216 srhmsubc 45250 |
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