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Theorem rngcid 41771
 Description: The identity arrow in the category of non-unital rings is the identity function. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 10-Mar-2020.)
Hypotheses
Ref Expression
rngccat.c 𝐶 = (RngCat‘𝑈)
rngcid.b 𝐵 = (Base‘𝐶)
rngcid.o 1 = (Id‘𝐶)
rngcid.u (𝜑𝑈𝑉)
rngcid.x (𝜑𝑋𝐵)
rngcid.s 𝑆 = (Base‘𝑋)
Assertion
Ref Expression
rngcid (𝜑 → ( 1𝑋) = ( I ↾ 𝑆))

Proof of Theorem rngcid
StepHypRef Expression
1 rngcid.o . . . 4 1 = (Id‘𝐶)
2 rngccat.c . . . . . 6 𝐶 = (RngCat‘𝑈)
3 rngcid.u . . . . . 6 (𝜑𝑈𝑉)
4 eqidd 2611 . . . . . 6 (𝜑 → (𝑈 ∩ Rng) = (𝑈 ∩ Rng))
5 eqidd 2611 . . . . . 6 (𝜑 → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) = ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))))
62, 3, 4, 5rngcval 41754 . . . . 5 (𝜑𝐶 = ((ExtStrCat‘𝑈) ↾cat ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))))
76fveq2d 6107 . . . 4 (𝜑 → (Id‘𝐶) = (Id‘((ExtStrCat‘𝑈) ↾cat ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))))))
81, 7syl5eq 2656 . . 3 (𝜑1 = (Id‘((ExtStrCat‘𝑈) ↾cat ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))))))
98fveq1d 6105 . 2 (𝜑 → ( 1𝑋) = ((Id‘((ExtStrCat‘𝑈) ↾cat ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))))‘𝑋))
10 eqid 2610 . . 3 ((ExtStrCat‘𝑈) ↾cat ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))) = ((ExtStrCat‘𝑈) ↾cat ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))))
11 eqid 2610 . . . 4 (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
12 incom 3767 . . . . 5 (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
1312a1i 11 . . . 4 (𝜑 → (𝑈 ∩ Rng) = (Rng ∩ 𝑈))
1411, 3, 13, 5rnghmsubcsetc 41769 . . 3 (𝜑 → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) ∈ (Subcat‘(ExtStrCat‘𝑈)))
154, 5rnghmresfn 41755 . . 3 (𝜑 → ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) Fn ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))
16 eqid 2610 . . 3 (Id‘(ExtStrCat‘𝑈)) = (Id‘(ExtStrCat‘𝑈))
17 rngcid.x . . . 4 (𝜑𝑋𝐵)
18 rngcid.b . . . . . 6 𝐵 = (Base‘𝐶)
192, 18, 3rngcbas 41757 . . . . 5 (𝜑𝐵 = (𝑈 ∩ Rng))
2019eleq2d 2673 . . . 4 (𝜑 → (𝑋𝐵𝑋 ∈ (𝑈 ∩ Rng)))
2117, 20mpbid 221 . . 3 (𝜑𝑋 ∈ (𝑈 ∩ Rng))
2210, 14, 15, 16, 21subcid 16330 . 2 (𝜑 → ((Id‘(ExtStrCat‘𝑈))‘𝑋) = ((Id‘((ExtStrCat‘𝑈) ↾cat ( RngHomo ↾ ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))))‘𝑋))
23 elinel1 3761 . . . . . 6 (𝑋 ∈ (𝑈 ∩ Rng) → 𝑋𝑈)
2420, 23syl6bi 242 . . . . 5 (𝜑 → (𝑋𝐵𝑋𝑈))
2517, 24mpd 15 . . . 4 (𝜑𝑋𝑈)
2611, 16, 3, 25estrcid 16597 . . 3 (𝜑 → ((Id‘(ExtStrCat‘𝑈))‘𝑋) = ( I ↾ (Base‘𝑋)))
27 rngcid.s . . . . . 6 𝑆 = (Base‘𝑋)
2827eqcomi 2619 . . . . 5 (Base‘𝑋) = 𝑆
2928a1i 11 . . . 4 (𝜑 → (Base‘𝑋) = 𝑆)
3029reseq2d 5317 . . 3 (𝜑 → ( I ↾ (Base‘𝑋)) = ( I ↾ 𝑆))
3126, 30eqtrd 2644 . 2 (𝜑 → ((Id‘(ExtStrCat‘𝑈))‘𝑋) = ( I ↾ 𝑆))
329, 22, 313eqtr2d 2650 1 (𝜑 → ( 1𝑋) = ( I ↾ 𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ∩ cin 3539   I cid 4948   × cxp 5036   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Idccid 16149   ↾cat cresc 16291  ExtStrCatcestrc 16585  Rngcrng 41664   RngHomo crngh 41675  RngCatcrngc 41749 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-hom 15793  df-cco 15794  df-0g 15925  df-cat 16152  df-cid 16153  df-homf 16154  df-ssc 16293  df-resc 16294  df-subc 16295  df-estrc 16586  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-grp 17248  df-ghm 17481  df-abl 18019  df-mgp 18313  df-mgmhm 41569  df-rng0 41665  df-rnghomo 41677  df-rngc 41751 This theorem is referenced by:  rngcsect  41772  rhmsubcrngclem1  41819  rhmsubclem3  41880
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