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Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmsubcALTVlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for rhmsubcALTV 44386. (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 2823 | . . 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 7191 | . . . 4 ⊢ (𝑥 GrpHom 𝑦) ∈ V | |
3 | 2 | inex1 5223 | . . 3 ⊢ ((𝑥 GrpHom 𝑦) ∩ ((mulGrp‘𝑥) MndHom (mulGrp‘𝑦))) ∈ V |
4 | 1, 3 | fnmpoi 7770 | . 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 19471 | . . . . . 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 5854 | . . . 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 4207 | . . . . . 6 ⊢ (Ring ∩ 𝑈) ⊆ Ring | |
12 | 10, 11 | eqsstrdi 4023 | . . . . 5 ⊢ (𝜑 → 𝑅 ⊆ Ring) |
13 | resmpo 7274 | . . . . 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 586 | . . . 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 2862 | . . 3 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((𝑥 GrpHom 𝑦) ∩ ((mulGrp‘𝑥) MndHom (mulGrp‘𝑦))))) |
16 | 15 | fneq1d 6448 | . 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 260 | 1 ⊢ (𝜑 → 𝐻 Fn (𝑅 × 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∩ cin 3937 ⊆ wss 3938 × cxp 5555 ↾ cres 5559 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 MndHom cmhm 17956 GrpHom cghm 18357 mulGrpcmgp 19241 Ringcrg 19299 RingHom crh 19466 RngCatALTVcrngcALTV 44236 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-plusg 16580 df-0g 16717 df-mhm 17958 df-ghm 18358 df-mgp 19242 df-ur 19254 df-ring 19301 df-rnghom 19469 |
This theorem is referenced by: rhmsubcALTV 44386 |
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