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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 18467 | . . 3 ⊢ Rel 𝐸 |
3 | 2 | a1i 11 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → Rel 𝐸) |
4 | subrgdvds.1 | . . . . . . . 8 ⊢ 𝑆 = (𝑅 ↾s 𝐴) | |
5 | 4 | subrgbas 18612 | . . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆)) |
6 | eqid 2610 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | 6 | subrgss 18604 | . . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅)) |
8 | 5, 7 | eqsstr3d 3603 | . . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) ⊆ (Base‘𝑅)) |
9 | 8 | sseld 3567 | . . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥 ∈ (Base‘𝑆) → 𝑥 ∈ (Base‘𝑅))) |
10 | eqid 2610 | . . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
11 | 4, 10 | ressmulr 15829 | . . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆)) |
12 | 11 | oveqd 6566 | . . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑧(.r‘𝑅)𝑥) = (𝑧(.r‘𝑆)𝑥)) |
13 | 12 | eqeq1d 2612 | . . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((𝑧(.r‘𝑅)𝑥) = 𝑦 ↔ (𝑧(.r‘𝑆)𝑥) = 𝑦)) |
14 | 13 | rexbidv 3034 | . . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑅)𝑥) = 𝑦 ↔ ∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦)) |
15 | ssrexv 3630 | . . . . . . 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 249 | . . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦 → ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)) |
18 | 9, 17 | anim12d 584 | . . . 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 2610 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
20 | eqid 2610 | . . . . 5 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
21 | 19, 1, 20 | dvdsr 18469 | . . . 4 ⊢ (𝑥𝐸𝑦 ↔ (𝑥 ∈ (Base‘𝑆) ∧ ∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦)) |
22 | subrgdvds.2 | . . . . 5 ⊢ ∥ = (∥r‘𝑅) | |
23 | 6, 22, 10 | dvdsr 18469 | . . . 4 ⊢ (𝑥 ∥ 𝑦 ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)) |
24 | 18, 21, 23 | 3imtr4g 284 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥𝐸𝑦 → 𝑥 ∥ 𝑦)) |
25 | df-br 4584 | . . 3 ⊢ (𝑥𝐸𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐸) | |
26 | df-br 4584 | . . 3 ⊢ (𝑥 ∥ 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ∥ ) | |
27 | 24, 25, 26 | 3imtr3g 283 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (〈𝑥, 𝑦〉 ∈ 𝐸 → 〈𝑥, 𝑦〉 ∈ ∥ )) |
28 | 3, 27 | relssdv 5135 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐸 ⊆ ∥ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ⊆ wss 3540 〈cop 4131 class class class wbr 4583 Rel wrel 5043 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 ↾s cress 15696 .rcmulr 15769 ∥rcdsr 18461 SubRingcsubrg 18599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-mulr 15782 df-subg 17414 df-ring 18372 df-dvdsr 18464 df-subrg 18601 |
This theorem is referenced by: subrguss 18618 |
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