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Mirrors > Home > MPE Home > Th. List > rhmeql | Structured version Visualization version GIF version |
Description: The equalizer of two ring homomorphisms is a subring. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
rhmeql | ⊢ ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝐺 ∈ (𝑆 RingHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubRing‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rhmghm 19477 | . . 3 ⊢ (𝐹 ∈ (𝑆 RingHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) | |
2 | rhmghm 19477 | . . 3 ⊢ (𝐺 ∈ (𝑆 RingHom 𝑇) → 𝐺 ∈ (𝑆 GrpHom 𝑇)) | |
3 | ghmeql 18381 | . . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubGrp‘𝑆)) | |
4 | 1, 2, 3 | syl2an 597 | . 2 ⊢ ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝐺 ∈ (𝑆 RingHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubGrp‘𝑆)) |
5 | eqid 2821 | . . . 4 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆) | |
6 | eqid 2821 | . . . 4 ⊢ (mulGrp‘𝑇) = (mulGrp‘𝑇) | |
7 | 5, 6 | rhmmhm 19474 | . . 3 ⊢ (𝐹 ∈ (𝑆 RingHom 𝑇) → 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇))) |
8 | 5, 6 | rhmmhm 19474 | . . 3 ⊢ (𝐺 ∈ (𝑆 RingHom 𝑇) → 𝐺 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇))) |
9 | mhmeql 17990 | . . 3 ⊢ ((𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)) ∧ 𝐺 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇))) → dom (𝐹 ∩ 𝐺) ∈ (SubMnd‘(mulGrp‘𝑆))) | |
10 | 7, 8, 9 | syl2an 597 | . 2 ⊢ ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝐺 ∈ (𝑆 RingHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubMnd‘(mulGrp‘𝑆))) |
11 | rhmrcl1 19471 | . . . 4 ⊢ (𝐹 ∈ (𝑆 RingHom 𝑇) → 𝑆 ∈ Ring) | |
12 | 11 | adantr 483 | . . 3 ⊢ ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝐺 ∈ (𝑆 RingHom 𝑇)) → 𝑆 ∈ Ring) |
13 | 5 | issubrg3 19563 | . . 3 ⊢ (𝑆 ∈ Ring → (dom (𝐹 ∩ 𝐺) ∈ (SubRing‘𝑆) ↔ (dom (𝐹 ∩ 𝐺) ∈ (SubGrp‘𝑆) ∧ dom (𝐹 ∩ 𝐺) ∈ (SubMnd‘(mulGrp‘𝑆))))) |
14 | 12, 13 | syl 17 | . 2 ⊢ ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝐺 ∈ (𝑆 RingHom 𝑇)) → (dom (𝐹 ∩ 𝐺) ∈ (SubRing‘𝑆) ↔ (dom (𝐹 ∩ 𝐺) ∈ (SubGrp‘𝑆) ∧ dom (𝐹 ∩ 𝐺) ∈ (SubMnd‘(mulGrp‘𝑆))))) |
15 | 4, 10, 14 | mpbir2and 711 | 1 ⊢ ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝐺 ∈ (𝑆 RingHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubRing‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 ∩ cin 3935 dom cdm 5555 ‘cfv 6355 (class class class)co 7156 MndHom cmhm 17954 SubMndcsubmnd 17955 SubGrpcsubg 18273 GrpHom cghm 18355 mulGrpcmgp 19239 Ringcrg 19297 RingHom crh 19464 SubRingcsubrg 19531 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-submnd 17957 df-grp 18106 df-minusg 18107 df-subg 18276 df-ghm 18356 df-mgp 19240 df-ur 19252 df-ring 19299 df-rnghom 19467 df-subrg 19533 |
This theorem is referenced by: evlseu 20296 |
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