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 44251 | . . . . . 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 498 | . . . 4 ⊢ (𝜑 → (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ ( I ↾ (Base‘𝑥)))) |
8 | 1, 7 | syl5eq 2866 | . . 3 ⊢ (𝜑 → 1 = (𝑥 ∈ 𝐵 ↦ ( I ↾ (Base‘𝑥)))) |
9 | fveq2 6663 | . . . . 5 ⊢ (𝑥 = 𝑋 → (Base‘𝑥) = (Base‘𝑋)) | |
10 | 9 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (Base‘𝑥) = (Base‘𝑋)) |
11 | 10 | reseq2d 5846 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ( I ↾ (Base‘𝑥)) = ( I ↾ (Base‘𝑋))) |
12 | rngcidALTV.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
13 | fvex 6676 | . . . 4 ⊢ (Base‘𝑋) ∈ V | |
14 | resiexg 7611 | . . . 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 6768 | . 2 ⊢ (𝜑 → ( 1 ‘𝑋) = ( I ↾ (Base‘𝑋))) |
17 | rngcidALTV.s | . . 3 ⊢ 𝑆 = (Base‘𝑋) | |
18 | 17 | reseq2i 5843 | . 2 ⊢ ( I ↾ 𝑆) = ( I ↾ (Base‘𝑋)) |
19 | 16, 18 | syl6eqr 2872 | 1 ⊢ (𝜑 → ( 1 ‘𝑋) = ( I ↾ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 Vcvv 3493 ↦ cmpt 5137 I cid 5452 ↾ cres 5550 ‘cfv 6348 Basecbs 16475 Catccat 16927 Idccid 16928 RngCatALTVcrngcALTV 44220 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-map 8400 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-fz 12885 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-hom 16581 df-cco 16582 df-0g 16707 df-cat 16931 df-cid 16932 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-ghm 18348 df-abl 18901 df-mgp 19232 df-mgmhm 44037 df-rng0 44137 df-rnghomo 44149 df-rngcALTV 44222 |
This theorem is referenced by: rngcsectALTV 44254 |
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