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Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmsubcALTVlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for rhmsubcALTV 42610. (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 |
---|---|
rhmsubcALTVlem1 | ⊢ (𝜑 → 𝐻 Fn (𝑅 × 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2752 | . . 3 ⊢ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((𝑥 GrpHom 𝑦) ∩ ((mulGrp‘𝑥) MndHom (mulGrp‘𝑦)))) = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((𝑥 GrpHom 𝑦) ∩ ((mulGrp‘𝑥) MndHom (mulGrp‘𝑦)))) | |
2 | ovex 6833 | . . . 4 ⊢ (𝑥 GrpHom 𝑦) ∈ V | |
3 | 2 | inex1 4943 | . . 3 ⊢ ((𝑥 GrpHom 𝑦) ∩ ((mulGrp‘𝑥) MndHom (mulGrp‘𝑦))) ∈ V |
4 | 1, 3 | fnmpt2i 7399 | . 2 ⊢ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((𝑥 GrpHom 𝑦) ∩ ((mulGrp‘𝑥) MndHom (mulGrp‘𝑦)))) Fn (𝑅 × 𝑅) |
5 | rngcrescrhmALTV.h | . . . . 5 ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) | |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))) |
7 | dfrhm2 18911 | . . . . . 6 ⊢ RingHom = (𝑥 ∈ Ring, 𝑦 ∈ Ring ↦ ((𝑥 GrpHom 𝑦) ∩ ((mulGrp‘𝑥) MndHom (mulGrp‘𝑦)))) | |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝜑 → RingHom = (𝑥 ∈ Ring, 𝑦 ∈ Ring ↦ ((𝑥 GrpHom 𝑦) ∩ ((mulGrp‘𝑥) MndHom (mulGrp‘𝑦))))) |
9 | 8 | reseq1d 5542 | . . . 4 ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) = ((𝑥 ∈ Ring, 𝑦 ∈ Ring ↦ ((𝑥 GrpHom 𝑦) ∩ ((mulGrp‘𝑥) MndHom (mulGrp‘𝑦)))) ↾ (𝑅 × 𝑅))) |
10 | rngcrescrhmALTV.r | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) | |
11 | inss1 3968 | . . . . . 6 ⊢ (Ring ∩ 𝑈) ⊆ Ring | |
12 | 10, 11 | syl6eqss 3788 | . . . . 5 ⊢ (𝜑 → 𝑅 ⊆ Ring) |
13 | resmpt2 6915 | . . . . 5 ⊢ ((𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring) → ((𝑥 ∈ Ring, 𝑦 ∈ Ring ↦ ((𝑥 GrpHom 𝑦) ∩ ((mulGrp‘𝑥) MndHom (mulGrp‘𝑦)))) ↾ (𝑅 × 𝑅)) = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((𝑥 GrpHom 𝑦) ∩ ((mulGrp‘𝑥) MndHom (mulGrp‘𝑦))))) | |
14 | 12, 12, 13 | syl2anc 696 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ Ring, 𝑦 ∈ Ring ↦ ((𝑥 GrpHom 𝑦) ∩ ((mulGrp‘𝑥) MndHom (mulGrp‘𝑦)))) ↾ (𝑅 × 𝑅)) = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((𝑥 GrpHom 𝑦) ∩ ((mulGrp‘𝑥) MndHom (mulGrp‘𝑦))))) |
15 | 6, 9, 14 | 3eqtrd 2790 | . . 3 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((𝑥 GrpHom 𝑦) ∩ ((mulGrp‘𝑥) MndHom (mulGrp‘𝑦))))) |
16 | 15 | fneq1d 6134 | . 2 ⊢ (𝜑 → (𝐻 Fn (𝑅 × 𝑅) ↔ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((𝑥 GrpHom 𝑦) ∩ ((mulGrp‘𝑥) MndHom (mulGrp‘𝑦)))) Fn (𝑅 × 𝑅))) |
17 | 4, 16 | mpbiri 248 | 1 ⊢ (𝜑 → 𝐻 Fn (𝑅 × 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1624 ∈ wcel 2131 ∩ cin 3706 ⊆ wss 3707 × cxp 5256 ↾ cres 5260 Fn wfn 6036 ‘cfv 6041 (class class class)co 6805 ↦ cmpt2 6807 MndHom cmhm 17526 GrpHom cghm 17850 mulGrpcmgp 18681 Ringcrg 18739 RingHom crh 18906 RngCatALTVcrngcALTV 42460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-rep 4915 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-cnex 10176 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-mulcom 10184 ax-addass 10185 ax-mulass 10186 ax-distr 10187 ax-i2m1 10188 ax-1ne0 10189 ax-1rid 10190 ax-rnegex 10191 ax-rrecex 10192 ax-cnre 10193 ax-pre-lttri 10194 ax-pre-lttrn 10195 ax-pre-ltadd 10196 ax-pre-mulgt0 10197 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4581 df-iun 4666 df-br 4797 df-opab 4857 df-mpt 4874 df-tr 4897 df-id 5166 df-eprel 5171 df-po 5179 df-so 5180 df-fr 5217 df-we 5219 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-pred 5833 df-ord 5879 df-on 5880 df-lim 5881 df-suc 5882 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-riota 6766 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-om 7223 df-1st 7325 df-2nd 7326 df-wrecs 7568 df-recs 7629 df-rdg 7667 df-er 7903 df-map 8017 df-en 8114 df-dom 8115 df-sdom 8116 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-sub 10452 df-neg 10453 df-nn 11205 df-2 11263 df-ndx 16054 df-slot 16055 df-base 16057 df-sets 16058 df-plusg 16148 df-0g 16296 df-mhm 17528 df-ghm 17851 df-mgp 18682 df-ur 18694 df-ring 18741 df-rnghom 18909 |
This theorem is referenced by: rhmsubcALTV 42610 |
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