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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 2824 | . . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
4 | 2, 3 | subrg1 19548 | . . . . . 6 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (1r‘𝑅) = (1r‘𝐻)) |
5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → (1r‘𝑅) = (1r‘𝐻)) |
6 | eqid 2824 | . . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
7 | 2, 6 | subrg0 19545 | . . . . . 6 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (0g‘𝑅) = (0g‘𝐻)) |
8 | 1, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → (0g‘𝑅) = (0g‘𝐻)) |
9 | 5, 8 | ifeq12d 4490 | . . . 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 5165 | . . 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 5165 | . 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 2824 | . . 3 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
14 | subrgmvr.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
15 | subrgrcl 19543 | . . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
16 | 1, 15 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) |
17 | 12, 13, 6, 3, 14, 16 | mvrfval 20203 | . 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 2824 | . . 3 ⊢ (𝐼 mVar 𝐻) = (𝐼 mVar 𝐻) | |
19 | eqid 2824 | . . 3 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
20 | eqid 2824 | . . 3 ⊢ (1r‘𝐻) = (1r‘𝐻) | |
21 | 2 | ovexi 7193 | . . . 4 ⊢ 𝐻 ∈ V |
22 | 21 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐻 ∈ V) |
23 | 18, 13, 19, 20, 14, 22 | mvrfval 20203 | . 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 2869 | 1 ⊢ (𝜑 → 𝑉 = (𝐼 mVar 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 {crab 3145 Vcvv 3497 ifcif 4470 ↦ cmpt 5149 ◡ccnv 5557 “ cima 5561 ‘cfv 6358 (class class class)co 7159 ↑m cmap 8409 Fincfn 8512 0cc0 10540 1c1 10541 ℕcn 11641 ℕ0cn0 11900 ↾s cress 16487 0gc0g 16716 1rcur 19254 Ringcrg 19300 SubRingcsubrg 19534 mVar cmvr 20135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-subg 18279 df-mgp 19243 df-ur 19255 df-ring 19302 df-subrg 19536 df-mvr 20140 |
This theorem is referenced by: subrgmvrf 20246 evlsvarsrng 20315 evlvar 20316 subrgvr1 20432 evls1var 20504 |
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