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| Mirrors > Home > MPE Home > Th. List > subrgdvds | Structured version Visualization version GIF version | ||
| Description: If an element divides another in a subring, then it also divides the other in the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
| Ref | Expression |
|---|---|
| subrgdvds.1 | ⊢ 𝑆 = (𝑅 ↾s 𝐴) |
| subrgdvds.2 | ⊢ ∥ = (∥r‘𝑅) |
| subrgdvds.3 | ⊢ 𝐸 = (∥r‘𝑆) |
| Ref | Expression |
|---|---|
| subrgdvds | ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐸 ⊆ ∥ ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | subrgdvds.3 | . . . 4 ⊢ 𝐸 = (∥r‘𝑆) | |
| 2 | 1 | reldvdsr 20360 | . . 3 ⊢ Rel 𝐸 |
| 3 | 2 | a1i 11 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → Rel 𝐸) |
| 4 | subrgdvds.1 | . . . . . . . 8 ⊢ 𝑆 = (𝑅 ↾s 𝐴) | |
| 5 | 4 | subrgbas 20581 | . . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆)) |
| 6 | eqid 2737 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 7 | 6 | subrgss 20572 | . . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅)) |
| 8 | 5, 7 | eqsstrrd 4019 | . . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) ⊆ (Base‘𝑅)) |
| 9 | 8 | sseld 3982 | . . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥 ∈ (Base‘𝑆) → 𝑥 ∈ (Base‘𝑅))) |
| 10 | eqid 2737 | . . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
| 11 | 4, 10 | ressmulr 17351 | . . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆)) |
| 12 | 11 | oveqd 7448 | . . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑧(.r‘𝑅)𝑥) = (𝑧(.r‘𝑆)𝑥)) |
| 13 | 12 | eqeq1d 2739 | . . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((𝑧(.r‘𝑅)𝑥) = 𝑦 ↔ (𝑧(.r‘𝑆)𝑥) = 𝑦)) |
| 14 | 13 | rexbidv 3179 | . . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑅)𝑥) = 𝑦 ↔ ∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦)) |
| 15 | ssrexv 4053 | . . . . . . 7 ⊢ ((Base‘𝑆) ⊆ (Base‘𝑅) → (∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑅)𝑥) = 𝑦 → ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)) | |
| 16 | 8, 15 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑅)𝑥) = 𝑦 → ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)) |
| 17 | 14, 16 | sylbird 260 | . . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦 → ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)) |
| 18 | 9, 17 | anim12d 609 | . . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((𝑥 ∈ (Base‘𝑆) ∧ ∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦) → (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦))) |
| 19 | eqid 2737 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
| 20 | eqid 2737 | . . . . 5 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
| 21 | 19, 1, 20 | dvdsr 20362 | . . . 4 ⊢ (𝑥𝐸𝑦 ↔ (𝑥 ∈ (Base‘𝑆) ∧ ∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦)) |
| 22 | subrgdvds.2 | . . . . 5 ⊢ ∥ = (∥r‘𝑅) | |
| 23 | 6, 22, 10 | dvdsr 20362 | . . . 4 ⊢ (𝑥 ∥ 𝑦 ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)) |
| 24 | 18, 21, 23 | 3imtr4g 296 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥𝐸𝑦 → 𝑥 ∥ 𝑦)) |
| 25 | df-br 5144 | . . 3 ⊢ (𝑥𝐸𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐸) | |
| 26 | df-br 5144 | . . 3 ⊢ (𝑥 ∥ 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ∥ ) | |
| 27 | 24, 25, 26 | 3imtr3g 295 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (〈𝑥, 𝑦〉 ∈ 𝐸 → 〈𝑥, 𝑦〉 ∈ ∥ )) |
| 28 | 3, 27 | relssdv 5798 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐸 ⊆ ∥ ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ⊆ wss 3951 〈cop 4632 class class class wbr 5143 Rel wrel 5690 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ↾s cress 17274 .rcmulr 17298 ∥rcdsr 20354 SubRingcsubrg 20569 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-mulr 17311 df-subg 19141 df-ring 20232 df-dvdsr 20357 df-subrg 20570 |
| This theorem is referenced by: subrguss 20587 |
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