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Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmsubcsetclem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for rhmsubcsetc 43031. (Contributed by AV, 9-Mar-2020.) |
Ref | Expression |
---|---|
rhmsubcsetc.c | ⊢ 𝐶 = (ExtStrCat‘𝑈) |
rhmsubcsetc.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
rhmsubcsetc.b | ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈)) |
rhmsubcsetc.h | ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) |
Ref | Expression |
---|---|
rhmsubcsetclem1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rhmsubcsetc.b | . . . . . 6 ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈)) | |
2 | 1 | eleq2d 2844 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Ring ∩ 𝑈))) |
3 | elin 4018 | . . . . . 6 ⊢ (𝑥 ∈ (Ring ∩ 𝑈) ↔ (𝑥 ∈ Ring ∧ 𝑥 ∈ 𝑈)) | |
4 | 3 | simplbi 493 | . . . . 5 ⊢ (𝑥 ∈ (Ring ∩ 𝑈) → 𝑥 ∈ Ring) |
5 | 2, 4 | syl6bi 245 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ Ring)) |
6 | 5 | imp 397 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ Ring) |
7 | eqid 2777 | . . . 4 ⊢ (Base‘𝑥) = (Base‘𝑥) | |
8 | 7 | idrhm 19120 | . . 3 ⊢ (𝑥 ∈ Ring → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥)) |
9 | 6, 8 | syl 17 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥)) |
10 | rhmsubcsetc.c | . . 3 ⊢ 𝐶 = (ExtStrCat‘𝑈) | |
11 | eqid 2777 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
12 | rhmsubcsetc.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
13 | 12 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑈 ∈ 𝑉) |
14 | 3 | simprbi 492 | . . . . 5 ⊢ (𝑥 ∈ (Ring ∩ 𝑈) → 𝑥 ∈ 𝑈) |
15 | 2, 14 | syl6bi 245 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝑈)) |
16 | 15 | imp 397 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝑈) |
17 | 10, 11, 13, 16 | estrcid 17159 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) = ( I ↾ (Base‘𝑥))) |
18 | rhmsubcsetc.h | . . . 4 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) | |
19 | 18 | oveqdr 6950 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥𝐻𝑥) = (𝑥( RingHom ↾ (𝐵 × 𝐵))𝑥)) |
20 | eqid 2777 | . . . . . . . 8 ⊢ (RingCat‘𝑈) = (RingCat‘𝑈) | |
21 | eqid 2777 | . . . . . . . 8 ⊢ (Base‘(RingCat‘𝑈)) = (Base‘(RingCat‘𝑈)) | |
22 | eqid 2777 | . . . . . . . 8 ⊢ (Hom ‘(RingCat‘𝑈)) = (Hom ‘(RingCat‘𝑈)) | |
23 | 20, 21, 12, 22 | ringchomfval 43020 | . . . . . . 7 ⊢ (𝜑 → (Hom ‘(RingCat‘𝑈)) = ( RingHom ↾ ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈))))) |
24 | 20, 21, 12 | ringcbas 43019 | . . . . . . . . . 10 ⊢ (𝜑 → (Base‘(RingCat‘𝑈)) = (𝑈 ∩ Ring)) |
25 | incom 4027 | . . . . . . . . . . . 12 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring) | |
26 | 1, 25 | syl6eq 2829 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) |
27 | 26 | eqcomd 2783 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑈 ∩ Ring) = 𝐵) |
28 | 24, 27 | eqtrd 2813 | . . . . . . . . 9 ⊢ (𝜑 → (Base‘(RingCat‘𝑈)) = 𝐵) |
29 | 28 | sqxpeqd 5387 | . . . . . . . 8 ⊢ (𝜑 → ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈))) = (𝐵 × 𝐵)) |
30 | 29 | reseq2d 5642 | . . . . . . 7 ⊢ (𝜑 → ( RingHom ↾ ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈)))) = ( RingHom ↾ (𝐵 × 𝐵))) |
31 | 23, 30 | eqtrd 2813 | . . . . . 6 ⊢ (𝜑 → (Hom ‘(RingCat‘𝑈)) = ( RingHom ↾ (𝐵 × 𝐵))) |
32 | 31 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (Hom ‘(RingCat‘𝑈)) = ( RingHom ↾ (𝐵 × 𝐵))) |
33 | 32 | eqcomd 2783 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( RingHom ↾ (𝐵 × 𝐵)) = (Hom ‘(RingCat‘𝑈))) |
34 | 33 | oveqd 6939 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥( RingHom ↾ (𝐵 × 𝐵))𝑥) = (𝑥(Hom ‘(RingCat‘𝑈))𝑥)) |
35 | 26 | eleq2d 2844 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (𝑈 ∩ Ring))) |
36 | 35 | biimpa 470 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝑈 ∩ Ring)) |
37 | 24 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (Base‘(RingCat‘𝑈)) = (𝑈 ∩ Ring)) |
38 | 36, 37 | eleqtrrd 2861 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (Base‘(RingCat‘𝑈))) |
39 | 20, 21, 13, 22, 38, 38 | ringchom 43021 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥(Hom ‘(RingCat‘𝑈))𝑥) = (𝑥 RingHom 𝑥)) |
40 | 19, 34, 39 | 3eqtrd 2817 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥𝐻𝑥) = (𝑥 RingHom 𝑥)) |
41 | 9, 17, 40 | 3eltr4d 2873 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ∩ cin 3790 I cid 5260 × cxp 5353 ↾ cres 5357 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 Hom chom 16349 Idccid 16711 ExtStrCatcestrc 17147 Ringcrg 18934 RingHom crh 19101 RingCatcringc 43011 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-hom 16362 df-cco 16363 df-0g 16488 df-cat 16714 df-cid 16715 df-resc 16856 df-estrc 17148 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-grp 17812 df-ghm 18042 df-mgp 18877 df-ur 18889 df-ring 18936 df-rnghom 19104 df-ringc 43013 |
This theorem is referenced by: rhmsubcsetc 43031 |
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