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Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmsubcsetclem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for rhmsubcsetc 44647. (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 2875 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Ring ∩ 𝑈))) |
3 | elin 3897 | . . . . . 6 ⊢ (𝑥 ∈ (Ring ∩ 𝑈) ↔ (𝑥 ∈ Ring ∧ 𝑥 ∈ 𝑈)) | |
4 | 3 | simplbi 501 | . . . . 5 ⊢ (𝑥 ∈ (Ring ∩ 𝑈) → 𝑥 ∈ Ring) |
5 | 2, 4 | syl6bi 256 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ Ring)) |
6 | 5 | imp 410 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ Ring) |
7 | eqid 2798 | . . . 4 ⊢ (Base‘𝑥) = (Base‘𝑥) | |
8 | 7 | idrhm 19479 | . . 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 2798 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
12 | rhmsubcsetc.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
13 | 12 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑈 ∈ 𝑉) |
14 | 3 | simprbi 500 | . . . . 5 ⊢ (𝑥 ∈ (Ring ∩ 𝑈) → 𝑥 ∈ 𝑈) |
15 | 2, 14 | syl6bi 256 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝑈)) |
16 | 15 | imp 410 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝑈) |
17 | 10, 11, 13, 16 | estrcid 17376 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) = ( I ↾ (Base‘𝑥))) |
18 | rhmsubcsetc.h | . . . 4 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) | |
19 | 18 | oveqdr 7163 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥𝐻𝑥) = (𝑥( RingHom ↾ (𝐵 × 𝐵))𝑥)) |
20 | eqid 2798 | . . . . . . . 8 ⊢ (RingCat‘𝑈) = (RingCat‘𝑈) | |
21 | eqid 2798 | . . . . . . . 8 ⊢ (Base‘(RingCat‘𝑈)) = (Base‘(RingCat‘𝑈)) | |
22 | eqid 2798 | . . . . . . . 8 ⊢ (Hom ‘(RingCat‘𝑈)) = (Hom ‘(RingCat‘𝑈)) | |
23 | 20, 21, 12, 22 | ringchomfval 44636 | . . . . . . 7 ⊢ (𝜑 → (Hom ‘(RingCat‘𝑈)) = ( RingHom ↾ ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈))))) |
24 | 20, 21, 12 | ringcbas 44635 | . . . . . . . . . 10 ⊢ (𝜑 → (Base‘(RingCat‘𝑈)) = (𝑈 ∩ Ring)) |
25 | incom 4128 | . . . . . . . . . . . 12 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring) | |
26 | 1, 25 | eqtrdi 2849 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) |
27 | 26 | eqcomd 2804 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑈 ∩ Ring) = 𝐵) |
28 | 24, 27 | eqtrd 2833 | . . . . . . . . 9 ⊢ (𝜑 → (Base‘(RingCat‘𝑈)) = 𝐵) |
29 | 28 | sqxpeqd 5551 | . . . . . . . 8 ⊢ (𝜑 → ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈))) = (𝐵 × 𝐵)) |
30 | 29 | reseq2d 5818 | . . . . . . 7 ⊢ (𝜑 → ( RingHom ↾ ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈)))) = ( RingHom ↾ (𝐵 × 𝐵))) |
31 | 23, 30 | eqtrd 2833 | . . . . . 6 ⊢ (𝜑 → (Hom ‘(RingCat‘𝑈)) = ( RingHom ↾ (𝐵 × 𝐵))) |
32 | 31 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (Hom ‘(RingCat‘𝑈)) = ( RingHom ↾ (𝐵 × 𝐵))) |
33 | 32 | eqcomd 2804 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( RingHom ↾ (𝐵 × 𝐵)) = (Hom ‘(RingCat‘𝑈))) |
34 | 33 | oveqd 7152 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥( RingHom ↾ (𝐵 × 𝐵))𝑥) = (𝑥(Hom ‘(RingCat‘𝑈))𝑥)) |
35 | 26 | eleq2d 2875 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (𝑈 ∩ Ring))) |
36 | 35 | biimpa 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝑈 ∩ Ring)) |
37 | 24 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (Base‘(RingCat‘𝑈)) = (𝑈 ∩ Ring)) |
38 | 36, 37 | eleqtrrd 2893 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (Base‘(RingCat‘𝑈))) |
39 | 20, 21, 13, 22, 38, 38 | ringchom 44637 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥(Hom ‘(RingCat‘𝑈))𝑥) = (𝑥 RingHom 𝑥)) |
40 | 19, 34, 39 | 3eqtrd 2837 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥𝐻𝑥) = (𝑥 RingHom 𝑥)) |
41 | 9, 17, 40 | 3eltr4d 2905 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∩ cin 3880 I cid 5424 × cxp 5517 ↾ cres 5521 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 Hom chom 16568 Idccid 16928 ExtStrCatcestrc 17364 Ringcrg 19290 RingHom crh 19460 RingCatcringc 44627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-hom 16581 df-cco 16582 df-0g 16707 df-cat 16931 df-cid 16932 df-resc 17073 df-estrc 17365 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-ghm 18348 df-mgp 19233 df-ur 19245 df-ring 19292 df-rnghom 19463 df-ringc 44629 |
This theorem is referenced by: rhmsubcsetc 44647 |
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