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Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmsubclem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for rhmsubc 45617. (Contributed by AV, 2-Mar-2020.) |
Ref | Expression |
---|---|
rngcrescrhm.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
rngcrescrhm.c | ⊢ 𝐶 = (RngCat‘𝑈) |
rngcrescrhm.r | ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) |
rngcrescrhm.h | ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) |
Ref | Expression |
---|---|
rhmsubclem1 | ⊢ (𝜑 → 𝐻 Fn (𝑅 × 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2740 | . . 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 7304 | . . . 4 ⊢ (𝑥 GrpHom 𝑦) ∈ V | |
3 | 2 | inex1 5245 | . . 3 ⊢ ((𝑥 GrpHom 𝑦) ∩ ((mulGrp‘𝑥) MndHom (mulGrp‘𝑦))) ∈ V |
4 | 1, 3 | fnmpoi 7903 | . 2 ⊢ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((𝑥 GrpHom 𝑦) ∩ ((mulGrp‘𝑥) MndHom (mulGrp‘𝑦)))) Fn (𝑅 × 𝑅) |
5 | rngcrescrhm.h | . . . . 5 ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) | |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))) |
7 | dfrhm2 19959 | . . . . . 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 5889 | . . . 4 ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) = ((𝑥 ∈ Ring, 𝑦 ∈ Ring ↦ ((𝑥 GrpHom 𝑦) ∩ ((mulGrp‘𝑥) MndHom (mulGrp‘𝑦)))) ↾ (𝑅 × 𝑅))) |
10 | rngcrescrhm.r | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) | |
11 | inss1 4168 | . . . . . 6 ⊢ (Ring ∩ 𝑈) ⊆ Ring | |
12 | 10, 11 | eqsstrdi 3980 | . . . . 5 ⊢ (𝜑 → 𝑅 ⊆ Ring) |
13 | resmpo 7388 | . . . . 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 584 | . . . 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 2784 | . . 3 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((𝑥 GrpHom 𝑦) ∩ ((mulGrp‘𝑥) MndHom (mulGrp‘𝑦))))) |
16 | 15 | fneq1d 6524 | . 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 257 | 1 ⊢ (𝜑 → 𝐻 Fn (𝑅 × 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 ∩ cin 3891 ⊆ wss 3892 × cxp 5588 ↾ cres 5592 Fn wfn 6427 ‘cfv 6432 (class class class)co 7271 ∈ cmpo 7273 MndHom cmhm 18426 GrpHom cghm 18829 mulGrpcmgp 19718 Ringcrg 19781 RingHom crh 19954 RngCatcrngc 45484 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-er 8481 df-map 8600 df-en 8717 df-dom 8718 df-sdom 8719 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-plusg 16973 df-0g 17150 df-mhm 18428 df-ghm 18830 df-mgp 19719 df-ur 19736 df-ring 19783 df-rnghom 19957 |
This theorem is referenced by: rhmsubc 45617 |
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