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Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmsubcsetclem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for rhmsubcsetc 47010. (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 2818 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Ring ∩ 𝑈))) |
3 | elin 3964 | . . . . . 6 ⊢ (𝑥 ∈ (Ring ∩ 𝑈) ↔ (𝑥 ∈ Ring ∧ 𝑥 ∈ 𝑈)) | |
4 | 3 | simplbi 497 | . . . . 5 ⊢ (𝑥 ∈ (Ring ∩ 𝑈) → 𝑥 ∈ Ring) |
5 | 2, 4 | syl6bi 253 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ Ring)) |
6 | 5 | imp 406 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ Ring) |
7 | eqid 2731 | . . . 4 ⊢ (Base‘𝑥) = (Base‘𝑥) | |
8 | 7 | idrhm 20382 | . . 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 2731 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
12 | rhmsubcsetc.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
13 | 12 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑈 ∈ 𝑉) |
14 | 3 | simprbi 496 | . . . . 5 ⊢ (𝑥 ∈ (Ring ∩ 𝑈) → 𝑥 ∈ 𝑈) |
15 | 2, 14 | syl6bi 253 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝑈)) |
16 | 15 | imp 406 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝑈) |
17 | 10, 11, 13, 16 | estrcid 18090 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) = ( I ↾ (Base‘𝑥))) |
18 | rhmsubcsetc.h | . . . 4 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) | |
19 | 18 | oveqdr 7440 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥𝐻𝑥) = (𝑥( RingHom ↾ (𝐵 × 𝐵))𝑥)) |
20 | eqid 2731 | . . . . . . . 8 ⊢ (RingCat‘𝑈) = (RingCat‘𝑈) | |
21 | eqid 2731 | . . . . . . . 8 ⊢ (Base‘(RingCat‘𝑈)) = (Base‘(RingCat‘𝑈)) | |
22 | eqid 2731 | . . . . . . . 8 ⊢ (Hom ‘(RingCat‘𝑈)) = (Hom ‘(RingCat‘𝑈)) | |
23 | 20, 21, 12, 22 | ringchomfval 46999 | . . . . . . 7 ⊢ (𝜑 → (Hom ‘(RingCat‘𝑈)) = ( RingHom ↾ ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈))))) |
24 | 20, 21, 12 | ringcbas 46998 | . . . . . . . . . 10 ⊢ (𝜑 → (Base‘(RingCat‘𝑈)) = (𝑈 ∩ Ring)) |
25 | incom 4201 | . . . . . . . . . . . 12 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring) | |
26 | 1, 25 | eqtrdi 2787 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) |
27 | 26 | eqcomd 2737 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑈 ∩ Ring) = 𝐵) |
28 | 24, 27 | eqtrd 2771 | . . . . . . . . 9 ⊢ (𝜑 → (Base‘(RingCat‘𝑈)) = 𝐵) |
29 | 28 | sqxpeqd 5708 | . . . . . . . 8 ⊢ (𝜑 → ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈))) = (𝐵 × 𝐵)) |
30 | 29 | reseq2d 5981 | . . . . . . 7 ⊢ (𝜑 → ( RingHom ↾ ((Base‘(RingCat‘𝑈)) × (Base‘(RingCat‘𝑈)))) = ( RingHom ↾ (𝐵 × 𝐵))) |
31 | 23, 30 | eqtrd 2771 | . . . . . 6 ⊢ (𝜑 → (Hom ‘(RingCat‘𝑈)) = ( RingHom ↾ (𝐵 × 𝐵))) |
32 | 31 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (Hom ‘(RingCat‘𝑈)) = ( RingHom ↾ (𝐵 × 𝐵))) |
33 | 32 | eqcomd 2737 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( RingHom ↾ (𝐵 × 𝐵)) = (Hom ‘(RingCat‘𝑈))) |
34 | 33 | oveqd 7429 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥( RingHom ↾ (𝐵 × 𝐵))𝑥) = (𝑥(Hom ‘(RingCat‘𝑈))𝑥)) |
35 | 26 | eleq2d 2818 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (𝑈 ∩ Ring))) |
36 | 35 | biimpa 476 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝑈 ∩ Ring)) |
37 | 24 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (Base‘(RingCat‘𝑈)) = (𝑈 ∩ Ring)) |
38 | 36, 37 | eleqtrrd 2835 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (Base‘(RingCat‘𝑈))) |
39 | 20, 21, 13, 22, 38, 38 | ringchom 47000 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥(Hom ‘(RingCat‘𝑈))𝑥) = (𝑥 RingHom 𝑥)) |
40 | 19, 34, 39 | 3eqtrd 2775 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥𝐻𝑥) = (𝑥 RingHom 𝑥)) |
41 | 9, 17, 40 | 3eltr4d 2847 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∩ cin 3947 I cid 5573 × cxp 5674 ↾ cres 5678 ‘cfv 6543 (class class class)co 7412 Basecbs 17149 Hom chom 17213 Idccid 17614 ExtStrCatcestrc 18078 Ringcrg 20128 RingHom crh 20361 RingCatcringc 46990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-fz 13490 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-hom 17226 df-cco 17227 df-0g 17392 df-cat 17617 df-cid 17618 df-resc 17763 df-estrc 18079 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18706 df-grp 18859 df-ghm 19129 df-mgp 20030 df-ur 20077 df-ring 20130 df-rhm 20364 df-ringc 46992 |
This theorem is referenced by: rhmsubcsetc 47010 |
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