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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 2772 | . . . . . 6 ⊢ (𝜑 → (𝑈 ∩ Ring) = (𝑈 ∩ Ring)) | |
5 | eqidd 2772 | . . . . . 6 ⊢ (𝜑 → ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring))) = ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))) | |
6 | 2, 3, 4, 5 | ringcval 43677 | . . . . 5 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring))))) |
7 | 6 | fveq2d 6500 | . . . 4 ⊢ (𝜑 → (Id‘𝐶) = (Id‘((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))))) |
8 | 1, 7 | syl5eq 2819 | . . 3 ⊢ (𝜑 → 1 = (Id‘((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))))) |
9 | 8 | fveq1d 6498 | . 2 ⊢ (𝜑 → ( 1 ‘𝑋) = ((Id‘((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))))‘𝑋)) |
10 | eqid 2771 | . . 3 ⊢ ((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))) = ((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))) | |
11 | eqid 2771 | . . . 4 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈) | |
12 | incom 4060 | . . . . 5 ⊢ (𝑈 ∩ Ring) = (Ring ∩ 𝑈) | |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑈 ∩ Ring) = (Ring ∩ 𝑈)) |
14 | 11, 3, 13, 5 | rhmsubcsetc 43692 | . . 3 ⊢ (𝜑 → ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring))) ∈ (Subcat‘(ExtStrCat‘𝑈))) |
15 | 4, 5 | rhmresfn 43678 | . . 3 ⊢ (𝜑 → ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring))) Fn ((𝑈 ∩ Ring) × (𝑈 ∩ Ring))) |
16 | eqid 2771 | . . 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 43680 | . . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) |
20 | 19 | eleq2d 2844 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ (𝑈 ∩ Ring))) |
21 | 17, 20 | mpbid 224 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑈 ∩ Ring)) |
22 | 10, 14, 15, 16, 21 | subcid 16987 | . 2 ⊢ (𝜑 → ((Id‘(ExtStrCat‘𝑈))‘𝑋) = ((Id‘((ExtStrCat‘𝑈) ↾cat ( RingHom ↾ ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))))‘𝑋)) |
23 | elinel1 4054 | . . . . . 6 ⊢ (𝑋 ∈ (𝑈 ∩ Ring) → 𝑋 ∈ 𝑈) | |
24 | 20, 23 | syl6bi 245 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝐵 → 𝑋 ∈ 𝑈)) |
25 | 17, 24 | mpd 15 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑈) |
26 | 11, 16, 3, 25 | estrcid 17254 | . . 3 ⊢ (𝜑 → ((Id‘(ExtStrCat‘𝑈))‘𝑋) = ( I ↾ (Base‘𝑋))) |
27 | ringcid.s | . . . . . 6 ⊢ 𝑆 = (Base‘𝑋) | |
28 | 27 | eqcomi 2780 | . . . . 5 ⊢ (Base‘𝑋) = 𝑆 |
29 | 28 | a1i 11 | . . . 4 ⊢ (𝜑 → (Base‘𝑋) = 𝑆) |
30 | 29 | reseq2d 5692 | . . 3 ⊢ (𝜑 → ( I ↾ (Base‘𝑋)) = ( I ↾ 𝑆)) |
31 | 26, 30 | eqtrd 2807 | . 2 ⊢ (𝜑 → ((Id‘(ExtStrCat‘𝑈))‘𝑋) = ( I ↾ 𝑆)) |
32 | 9, 22, 31 | 3eqtr2d 2813 | 1 ⊢ (𝜑 → ( 1 ‘𝑋) = ( I ↾ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 ∈ wcel 2051 ∩ cin 3821 I cid 5307 × cxp 5401 ↾ cres 5405 ‘cfv 6185 (class class class)co 6974 Basecbs 16337 Idccid 16806 ↾cat cresc 16948 ExtStrCatcestrc 17242 Ringcrg 19032 RingHom crh 19199 RingCatcringc 43672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-map 8206 df-pm 8207 df-ixp 8258 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-z 11792 df-dec 11910 df-uz 12057 df-fz 12707 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-hom 16443 df-cco 16444 df-0g 16569 df-cat 16809 df-cid 16810 df-homf 16811 df-ssc 16950 df-resc 16951 df-subc 16952 df-estrc 17243 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-mhm 17815 df-grp 17906 df-ghm 18139 df-mgp 18975 df-ur 18987 df-ring 19034 df-rnghom 19202 df-ringc 43674 |
This theorem is referenced by: ringcsect 43700 funcringcsetcALTV2lem7 43711 srhmsubc 43745 |
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