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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 19396 | . . 3 ⊢ Rel 𝐸 |
3 | 2 | a1i 11 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → Rel 𝐸) |
4 | subrgdvds.1 | . . . . . . . 8 ⊢ 𝑆 = (𝑅 ↾s 𝐴) | |
5 | 4 | subrgbas 19546 | . . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆)) |
6 | eqid 2823 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | 6 | subrgss 19538 | . . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅)) |
8 | 5, 7 | eqsstrrd 4008 | . . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) ⊆ (Base‘𝑅)) |
9 | 8 | sseld 3968 | . . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥 ∈ (Base‘𝑆) → 𝑥 ∈ (Base‘𝑅))) |
10 | eqid 2823 | . . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
11 | 4, 10 | ressmulr 16627 | . . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆)) |
12 | 11 | oveqd 7175 | . . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑧(.r‘𝑅)𝑥) = (𝑧(.r‘𝑆)𝑥)) |
13 | 12 | eqeq1d 2825 | . . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((𝑧(.r‘𝑅)𝑥) = 𝑦 ↔ (𝑧(.r‘𝑆)𝑥) = 𝑦)) |
14 | 13 | rexbidv 3299 | . . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑅)𝑥) = 𝑦 ↔ ∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦)) |
15 | ssrexv 4036 | . . . . . . 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 262 | . . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦 → ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)) |
18 | 9, 17 | anim12d 610 | . . . 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 2823 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
20 | eqid 2823 | . . . . 5 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
21 | 19, 1, 20 | dvdsr 19398 | . . . 4 ⊢ (𝑥𝐸𝑦 ↔ (𝑥 ∈ (Base‘𝑆) ∧ ∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦)) |
22 | subrgdvds.2 | . . . . 5 ⊢ ∥ = (∥r‘𝑅) | |
23 | 6, 22, 10 | dvdsr 19398 | . . . 4 ⊢ (𝑥 ∥ 𝑦 ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)) |
24 | 18, 21, 23 | 3imtr4g 298 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥𝐸𝑦 → 𝑥 ∥ 𝑦)) |
25 | df-br 5069 | . . 3 ⊢ (𝑥𝐸𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐸) | |
26 | df-br 5069 | . . 3 ⊢ (𝑥 ∥ 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ∥ ) | |
27 | 24, 25, 26 | 3imtr3g 297 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (〈𝑥, 𝑦〉 ∈ 𝐸 → 〈𝑥, 𝑦〉 ∈ ∥ )) |
28 | 3, 27 | relssdv 5663 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐸 ⊆ ∥ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 ⊆ wss 3938 〈cop 4575 class class class wbr 5068 Rel wrel 5562 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 ↾s cress 16486 .rcmulr 16568 ∥rcdsr 19390 SubRingcsubrg 19533 |
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-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 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-3 11704 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-mulr 16581 df-subg 18278 df-ring 19301 df-dvdsr 19393 df-subrg 19535 |
This theorem is referenced by: subrguss 19552 |
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