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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 20377 | . . 3 ⊢ Rel 𝐸 |
3 | 2 | a1i 11 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → Rel 𝐸) |
4 | subrgdvds.1 | . . . . . . . 8 ⊢ 𝑆 = (𝑅 ↾s 𝐴) | |
5 | 4 | subrgbas 20598 | . . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆)) |
6 | eqid 2735 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | 6 | subrgss 20589 | . . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅)) |
8 | 5, 7 | eqsstrrd 4035 | . . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) ⊆ (Base‘𝑅)) |
9 | 8 | sseld 3994 | . . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥 ∈ (Base‘𝑆) → 𝑥 ∈ (Base‘𝑅))) |
10 | eqid 2735 | . . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
11 | 4, 10 | ressmulr 17353 | . . . . . . . . 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 2737 | . . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((𝑧(.r‘𝑅)𝑥) = 𝑦 ↔ (𝑧(.r‘𝑆)𝑥) = 𝑦)) |
14 | 13 | rexbidv 3177 | . . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑅)𝑥) = 𝑦 ↔ ∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦)) |
15 | ssrexv 4065 | . . . . . . 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 2735 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
20 | eqid 2735 | . . . . 5 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
21 | 19, 1, 20 | dvdsr 20379 | . . . 4 ⊢ (𝑥𝐸𝑦 ↔ (𝑥 ∈ (Base‘𝑆) ∧ ∃𝑧 ∈ (Base‘𝑆)(𝑧(.r‘𝑆)𝑥) = 𝑦)) |
22 | subrgdvds.2 | . . . . 5 ⊢ ∥ = (∥r‘𝑅) | |
23 | 6, 22, 10 | dvdsr 20379 | . . . 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 5149 | . . 3 ⊢ (𝑥𝐸𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐸) | |
26 | df-br 5149 | . . 3 ⊢ (𝑥 ∥ 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ∥ ) | |
27 | 24, 25, 26 | 3imtr3g 295 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (〈𝑥, 𝑦〉 ∈ 𝐸 → 〈𝑥, 𝑦〉 ∈ ∥ )) |
28 | 3, 27 | relssdv 5801 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐸 ⊆ ∥ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 ⊆ wss 3963 〈cop 4637 class class class wbr 5148 Rel wrel 5694 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 ↾s cress 17274 .rcmulr 17299 ∥rcdsr 20371 SubRingcsubrg 20586 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-mulr 17312 df-subg 19154 df-ring 20253 df-dvdsr 20374 df-subrg 20587 |
This theorem is referenced by: subrguss 20604 |
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