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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 19388 | . . 3 ⊢ Rel 𝐸 |
3 | 2 | a1i 11 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → Rel 𝐸) |
4 | subrgdvds.1 | . . . . . . . 8 ⊢ 𝑆 = (𝑅 ↾s 𝐴) | |
5 | 4 | subrgbas 19538 | . . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆)) |
6 | eqid 2821 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | 6 | subrgss 19530 | . . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅)) |
8 | 5, 7 | eqsstrrd 4006 | . . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) ⊆ (Base‘𝑅)) |
9 | 8 | sseld 3966 | . . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥 ∈ (Base‘𝑆) → 𝑥 ∈ (Base‘𝑅))) |
10 | eqid 2821 | . . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
11 | 4, 10 | ressmulr 16619 | . . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆)) |
12 | 11 | oveqd 7167 | . . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑧(.r‘𝑅)𝑥) = (𝑧(.r‘𝑆)𝑥)) |
13 | 12 | eqeq1d 2823 | . . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((𝑧(.r‘𝑅)𝑥) = 𝑦 ↔ (𝑧(.r‘𝑆)𝑥) = 𝑦)) |
14 | 13 | rexbidv 3297 | . . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑅)𝑥) = 𝑦 ↔ ∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦)) |
15 | ssrexv 4034 | . . . . . . 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 2821 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
20 | eqid 2821 | . . . . 5 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
21 | 19, 1, 20 | dvdsr 19390 | . . . 4 ⊢ (𝑥𝐸𝑦 ↔ (𝑥 ∈ (Base‘𝑆) ∧ ∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦)) |
22 | subrgdvds.2 | . . . . 5 ⊢ ∥ = (∥r‘𝑅) | |
23 | 6, 22, 10 | dvdsr 19390 | . . . 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 5060 | . . 3 ⊢ (𝑥𝐸𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐸) | |
26 | df-br 5060 | . . 3 ⊢ (𝑥 ∥ 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ∥ ) | |
27 | 24, 25, 26 | 3imtr3g 297 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (〈𝑥, 𝑦〉 ∈ 𝐸 → 〈𝑥, 𝑦〉 ∈ ∥ )) |
28 | 3, 27 | relssdv 5656 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐸 ⊆ ∥ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 ⊆ wss 3936 〈cop 4567 class class class wbr 5059 Rel wrel 5555 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 ↾s cress 16478 .rcmulr 16560 ∥rcdsr 19382 SubRingcsubrg 19525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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-rab 3147 df-v 3497 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 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-mulr 16573 df-subg 18270 df-ring 19293 df-dvdsr 19385 df-subrg 19527 |
This theorem is referenced by: subrguss 19544 |
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