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Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmsubcALTVlem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for rhmsubcALTV 45293. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rngcrescrhmALTV.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
rngcrescrhmALTV.c | ⊢ 𝐶 = (RngCatALTV‘𝑈) |
rngcrescrhmALTV.r | ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) |
rngcrescrhmALTV.h | ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) |
Ref | Expression |
---|---|
rhmsubcALTVlem3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → ((Id‘(RngCatALTV‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngcrescrhmALTV.r | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) | |
2 | 1 | eleq2d 2819 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑅 ↔ 𝑥 ∈ (Ring ∩ 𝑈))) |
3 | elinel1 4099 | . . . . 5 ⊢ (𝑥 ∈ (Ring ∩ 𝑈) → 𝑥 ∈ Ring) | |
4 | 2, 3 | syl6bi 256 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑅 → 𝑥 ∈ Ring)) |
5 | 4 | imp 410 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → 𝑥 ∈ Ring) |
6 | eqid 2734 | . . . 4 ⊢ (Base‘𝑥) = (Base‘𝑥) | |
7 | 6 | idrhm 19723 | . . 3 ⊢ (𝑥 ∈ Ring → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥)) |
8 | 5, 7 | syl 17 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥)) |
9 | rngcrescrhmALTV.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
10 | 9 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → 𝑈 ∈ 𝑉) |
11 | eqid 2734 | . . . . 5 ⊢ (RngCatALTV‘𝑈) = (RngCatALTV‘𝑈) | |
12 | eqid 2734 | . . . . 5 ⊢ (Base‘(RngCatALTV‘𝑈)) = (Base‘(RngCatALTV‘𝑈)) | |
13 | 11, 12 | rngccatidALTV 45174 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → ((RngCatALTV‘𝑈) ∈ Cat ∧ (Id‘(RngCatALTV‘𝑈)) = (𝑦 ∈ (Base‘(RngCatALTV‘𝑈)) ↦ ( I ↾ (Base‘𝑦))))) |
14 | simpr 488 | . . . 4 ⊢ (((RngCatALTV‘𝑈) ∈ Cat ∧ (Id‘(RngCatALTV‘𝑈)) = (𝑦 ∈ (Base‘(RngCatALTV‘𝑈)) ↦ ( I ↾ (Base‘𝑦)))) → (Id‘(RngCatALTV‘𝑈)) = (𝑦 ∈ (Base‘(RngCatALTV‘𝑈)) ↦ ( I ↾ (Base‘𝑦)))) | |
15 | 10, 13, 14 | 3syl 18 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → (Id‘(RngCatALTV‘𝑈)) = (𝑦 ∈ (Base‘(RngCatALTV‘𝑈)) ↦ ( I ↾ (Base‘𝑦)))) |
16 | fveq2 6706 | . . . . 5 ⊢ (𝑦 = 𝑥 → (Base‘𝑦) = (Base‘𝑥)) | |
17 | 16 | reseq2d 5840 | . . . 4 ⊢ (𝑦 = 𝑥 → ( I ↾ (Base‘𝑦)) = ( I ↾ (Base‘𝑥))) |
18 | 17 | adantl 485 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑅) ∧ 𝑦 = 𝑥) → ( I ↾ (Base‘𝑦)) = ( I ↾ (Base‘𝑥))) |
19 | incom 4105 | . . . . . . . 8 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring) | |
20 | 1, 19 | eqtrdi 2790 | . . . . . . 7 ⊢ (𝜑 → 𝑅 = (𝑈 ∩ Ring)) |
21 | 20 | eleq2d 2819 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑅 ↔ 𝑥 ∈ (𝑈 ∩ Ring))) |
22 | ringrng 45064 | . . . . . . . 8 ⊢ (𝑥 ∈ Ring → 𝑥 ∈ Rng) | |
23 | 22 | anim2i 620 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝑈 ∧ 𝑥 ∈ Ring) → (𝑥 ∈ 𝑈 ∧ 𝑥 ∈ Rng)) |
24 | elin 3873 | . . . . . . 7 ⊢ (𝑥 ∈ (𝑈 ∩ Ring) ↔ (𝑥 ∈ 𝑈 ∧ 𝑥 ∈ Ring)) | |
25 | elin 3873 | . . . . . . 7 ⊢ (𝑥 ∈ (𝑈 ∩ Rng) ↔ (𝑥 ∈ 𝑈 ∧ 𝑥 ∈ Rng)) | |
26 | 23, 24, 25 | 3imtr4i 295 | . . . . . 6 ⊢ (𝑥 ∈ (𝑈 ∩ Ring) → 𝑥 ∈ (𝑈 ∩ Rng)) |
27 | 21, 26 | syl6bi 256 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑅 → 𝑥 ∈ (𝑈 ∩ Rng))) |
28 | 27 | imp 410 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → 𝑥 ∈ (𝑈 ∩ Rng)) |
29 | rngcrescrhmALTV.c | . . . . . 6 ⊢ 𝐶 = (RngCatALTV‘𝑈) | |
30 | 29 | eqcomi 2743 | . . . . . . 7 ⊢ (RngCatALTV‘𝑈) = 𝐶 |
31 | 30 | fveq2i 6709 | . . . . . 6 ⊢ (Base‘(RngCatALTV‘𝑈)) = (Base‘𝐶) |
32 | 29, 31, 9 | rngcbasALTV 45168 | . . . . 5 ⊢ (𝜑 → (Base‘(RngCatALTV‘𝑈)) = (𝑈 ∩ Rng)) |
33 | 32 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → (Base‘(RngCatALTV‘𝑈)) = (𝑈 ∩ Rng)) |
34 | 28, 33 | eleqtrrd 2837 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → 𝑥 ∈ (Base‘(RngCatALTV‘𝑈))) |
35 | fvexd 6721 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → (Base‘𝑥) ∈ V) | |
36 | 35 | resiexd 7021 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → ( I ↾ (Base‘𝑥)) ∈ V) |
37 | 15, 18, 34, 36 | fvmptd 6814 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → ((Id‘(RngCatALTV‘𝑈))‘𝑥) = ( I ↾ (Base‘𝑥))) |
38 | rngcrescrhmALTV.h | . . . 4 ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) | |
39 | 9, 29, 1, 38 | rhmsubcALTVlem2 45290 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅 ∧ 𝑥 ∈ 𝑅) → (𝑥𝐻𝑥) = (𝑥 RingHom 𝑥)) |
40 | 39 | 3anidm23 1423 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → (𝑥𝐻𝑥) = (𝑥 RingHom 𝑥)) |
41 | 8, 37, 40 | 3eltr4d 2849 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → ((Id‘(RngCatALTV‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3401 ∩ cin 3856 ↦ cmpt 5124 I cid 5443 × cxp 5538 ↾ cres 5542 ‘cfv 6369 (class class class)co 7202 Basecbs 16684 Catccat 17139 Idccid 17140 Ringcrg 19534 RingHom crh 19704 Rngcrng 45059 RngCatALTVcrngcALTV 45143 |
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 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
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 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-fz 13079 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-plusg 16780 df-hom 16791 df-cco 16792 df-0g 16918 df-cat 17143 df-cid 17144 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-mhm 18190 df-grp 18340 df-minusg 18341 df-ghm 18592 df-cmn 19144 df-abl 19145 df-mgp 19477 df-ur 19489 df-ring 19536 df-rnghom 19707 df-mgmhm 44960 df-rng0 45060 df-rnghomo 45072 df-rngcALTV 45145 |
This theorem is referenced by: rhmsubcALTV 45293 |
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