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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngcidALTV | Structured version Visualization version GIF version |
Description: The identity arrow in the category of non-unital rings is the identity function. (Contributed by AV, 27-Feb-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rngccatALTV.c | ⊢ 𝐶 = (RngCatALTV‘𝑈) |
rngcidALTV.b | ⊢ 𝐵 = (Base‘𝐶) |
rngcidALTV.o | ⊢ 1 = (Id‘𝐶) |
rngcidALTV.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
rngcidALTV.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
rngcidALTV.s | ⊢ 𝑆 = (Base‘𝑋) |
Ref | Expression |
---|---|
rngcidALTV | ⊢ (𝜑 → ( 1 ‘𝑋) = ( I ↾ 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngcidALTV.o | . . . 4 ⊢ 1 = (Id‘𝐶) | |
2 | rngcidALTV.u | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
3 | rngccatALTV.c | . . . . . . 7 ⊢ 𝐶 = (RngCatALTV‘𝑈) | |
4 | rngcidALTV.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐶) | |
5 | 3, 4 | rngccatidALTV 43765 | . . . . . 6 ⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ ( I ↾ (Base‘𝑥))))) |
6 | 2, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ ( I ↾ (Base‘𝑥))))) |
7 | 6 | simprd 496 | . . . 4 ⊢ (𝜑 → (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ ( I ↾ (Base‘𝑥)))) |
8 | 1, 7 | syl5eq 2843 | . . 3 ⊢ (𝜑 → 1 = (𝑥 ∈ 𝐵 ↦ ( I ↾ (Base‘𝑥)))) |
9 | fveq2 6543 | . . . . 5 ⊢ (𝑥 = 𝑋 → (Base‘𝑥) = (Base‘𝑋)) | |
10 | 9 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (Base‘𝑥) = (Base‘𝑋)) |
11 | 10 | reseq2d 5739 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ( I ↾ (Base‘𝑥)) = ( I ↾ (Base‘𝑋))) |
12 | rngcidALTV.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
13 | fvex 6556 | . . . 4 ⊢ (Base‘𝑋) ∈ V | |
14 | resiexg 7480 | . . . 4 ⊢ ((Base‘𝑋) ∈ V → ( I ↾ (Base‘𝑋)) ∈ V) | |
15 | 13, 14 | mp1i 13 | . . 3 ⊢ (𝜑 → ( I ↾ (Base‘𝑋)) ∈ V) |
16 | 8, 11, 12, 15 | fvmptd 6646 | . 2 ⊢ (𝜑 → ( 1 ‘𝑋) = ( I ↾ (Base‘𝑋))) |
17 | rngcidALTV.s | . . 3 ⊢ 𝑆 = (Base‘𝑋) | |
18 | 17 | reseq2i 5736 | . 2 ⊢ ( I ↾ 𝑆) = ( I ↾ (Base‘𝑋)) |
19 | 16, 18 | syl6eqr 2849 | 1 ⊢ (𝜑 → ( 1 ‘𝑋) = ( I ↾ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 Vcvv 3437 ↦ cmpt 5045 I cid 5352 ↾ cres 5450 ‘cfv 6230 Basecbs 16317 Catccat 16769 Idccid 16770 RngCatALTVcrngcALTV 43734 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-om 7442 df-1st 7550 df-2nd 7551 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-1o 7958 df-oadd 7962 df-er 8144 df-map 8263 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-nn 11492 df-2 11553 df-3 11554 df-4 11555 df-5 11556 df-6 11557 df-7 11558 df-8 11559 df-9 11560 df-n0 11751 df-z 11835 df-dec 11953 df-uz 12099 df-fz 12748 df-struct 16319 df-ndx 16320 df-slot 16321 df-base 16323 df-sets 16324 df-plusg 16412 df-hom 16423 df-cco 16424 df-0g 16549 df-cat 16773 df-cid 16774 df-mgm 17686 df-sgrp 17728 df-mnd 17739 df-mhm 17779 df-grp 17869 df-ghm 18102 df-abl 18641 df-mgp 18935 df-mgmhm 43555 df-rng0 43651 df-rnghomo 43663 df-rngcALTV 43736 |
This theorem is referenced by: rngcsectALTV 43768 |
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