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Mirrors > Home > MPE Home > Th. List > subrgmvr | Structured version Visualization version GIF version |
Description: The variables in a subring polynomial algebra are the same as the original ring. (Contributed by Mario Carneiro, 4-Jul-2015.) |
Ref | Expression |
---|---|
subrgmvr.v | ⊢ 𝑉 = (𝐼 mVar 𝑅) |
subrgmvr.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
subrgmvr.r | ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) |
subrgmvr.h | ⊢ 𝐻 = (𝑅 ↾s 𝑇) |
Ref | Expression |
---|---|
subrgmvr | ⊢ (𝜑 → 𝑉 = (𝐼 mVar 𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subrgmvr.r | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
2 | subrgmvr.h | . . . . . . 7 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
3 | eqid 2798 | . . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
4 | 2, 3 | subrg1 19538 | . . . . . 6 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (1r‘𝑅) = (1r‘𝐻)) |
5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → (1r‘𝑅) = (1r‘𝐻)) |
6 | eqid 2798 | . . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
7 | 2, 6 | subrg0 19535 | . . . . . 6 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (0g‘𝑅) = (0g‘𝐻)) |
8 | 1, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → (0g‘𝑅) = (0g‘𝐻)) |
9 | 5, 8 | ifeq12d 4445 | . . . 4 ⊢ (𝜑 → if(𝑦 = (𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, 1, 0)), (1r‘𝑅), (0g‘𝑅)) = if(𝑦 = (𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, 1, 0)), (1r‘𝐻), (0g‘𝐻))) |
10 | 9 | mpteq2dv 5126 | . . 3 ⊢ (𝜑 → (𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = (𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, 1, 0)), (1r‘𝑅), (0g‘𝑅))) = (𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = (𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, 1, 0)), (1r‘𝐻), (0g‘𝐻)))) |
11 | 10 | mpteq2dv 5126 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = (𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, 1, 0)), (1r‘𝑅), (0g‘𝑅)))) = (𝑥 ∈ 𝐼 ↦ (𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = (𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, 1, 0)), (1r‘𝐻), (0g‘𝐻))))) |
12 | subrgmvr.v | . . 3 ⊢ 𝑉 = (𝐼 mVar 𝑅) | |
13 | eqid 2798 | . . 3 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
14 | subrgmvr.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
15 | subrgrcl 19533 | . . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
16 | 1, 15 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) |
17 | 12, 13, 6, 3, 14, 16 | mvrfval 20658 | . 2 ⊢ (𝜑 → 𝑉 = (𝑥 ∈ 𝐼 ↦ (𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = (𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, 1, 0)), (1r‘𝑅), (0g‘𝑅))))) |
18 | eqid 2798 | . . 3 ⊢ (𝐼 mVar 𝐻) = (𝐼 mVar 𝐻) | |
19 | eqid 2798 | . . 3 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
20 | eqid 2798 | . . 3 ⊢ (1r‘𝐻) = (1r‘𝐻) | |
21 | 2 | ovexi 7169 | . . . 4 ⊢ 𝐻 ∈ V |
22 | 21 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐻 ∈ V) |
23 | 18, 13, 19, 20, 14, 22 | mvrfval 20658 | . 2 ⊢ (𝜑 → (𝐼 mVar 𝐻) = (𝑥 ∈ 𝐼 ↦ (𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = (𝑧 ∈ 𝐼 ↦ if(𝑧 = 𝑥, 1, 0)), (1r‘𝐻), (0g‘𝐻))))) |
24 | 11, 17, 23 | 3eqtr4d 2843 | 1 ⊢ (𝜑 → 𝑉 = (𝐼 mVar 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 ifcif 4425 ↦ cmpt 5110 ◡ccnv 5518 “ cima 5522 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 Fincfn 8492 0cc0 10526 1c1 10527 ℕcn 11625 ℕ0cn0 11885 ↾s cress 16476 0gc0g 16705 1rcur 19244 Ringcrg 19290 SubRingcsubrg 19524 mVar cmvr 20590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-subg 18268 df-mgp 19233 df-ur 19245 df-ring 19292 df-subrg 19526 df-mvr 20595 |
This theorem is referenced by: subrgmvrf 20702 evlsvarsrng 20771 evlvar 20772 subrgvr1 20890 evls1var 20962 |
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