Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > subrgmvrf | Structured version Visualization version GIF version |
Description: The variables in a polynomial algebra are contained in every subring algebra. (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) |
subrgmvrf.u | ⊢ 𝑈 = (𝐼 mPoly 𝐻) |
subrgmvrf.b | ⊢ 𝐵 = (Base‘𝑈) |
Ref | Expression |
---|---|
subrgmvrf | ⊢ (𝜑 → 𝑉:𝐼⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
2 | subrgmvr.v | . . . 4 ⊢ 𝑉 = (𝐼 mVar 𝑅) | |
3 | eqid 2610 | . . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅)) | |
4 | subrgmvr.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
5 | subrgmvr.r | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
6 | subrgrcl 18608 | . . . . 5 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) |
8 | 1, 2, 3, 4, 7 | mvrf 19245 | . . 3 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘(𝐼 mPwSer 𝑅))) |
9 | ffn 5958 | . . 3 ⊢ (𝑉:𝐼⟶(Base‘(𝐼 mPwSer 𝑅)) → 𝑉 Fn 𝐼) | |
10 | 8, 9 | syl 17 | . 2 ⊢ (𝜑 → 𝑉 Fn 𝐼) |
11 | subrgmvr.h | . . . . . . 7 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
12 | 2, 4, 5, 11 | subrgmvr 19282 | . . . . . 6 ⊢ (𝜑 → 𝑉 = (𝐼 mVar 𝐻)) |
13 | 12 | fveq1d 6105 | . . . . 5 ⊢ (𝜑 → (𝑉‘𝑥) = ((𝐼 mVar 𝐻)‘𝑥)) |
14 | 13 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑉‘𝑥) = ((𝐼 mVar 𝐻)‘𝑥)) |
15 | subrgmvrf.u | . . . . 5 ⊢ 𝑈 = (𝐼 mPoly 𝐻) | |
16 | eqid 2610 | . . . . 5 ⊢ (𝐼 mVar 𝐻) = (𝐼 mVar 𝐻) | |
17 | subrgmvrf.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑈) | |
18 | 4 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ 𝑊) |
19 | 11 | subrgring 18606 | . . . . . . 7 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐻 ∈ Ring) |
20 | 5, 19 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐻 ∈ Ring) |
21 | 20 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐻 ∈ Ring) |
22 | simpr 476 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼) | |
23 | 15, 16, 17, 18, 21, 22 | mvrcl 19270 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐼 mVar 𝐻)‘𝑥) ∈ 𝐵) |
24 | 14, 23 | eqeltrd 2688 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑉‘𝑥) ∈ 𝐵) |
25 | 24 | ralrimiva 2949 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝑉‘𝑥) ∈ 𝐵) |
26 | ffnfv 6295 | . 2 ⊢ (𝑉:𝐼⟶𝐵 ↔ (𝑉 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑉‘𝑥) ∈ 𝐵)) | |
27 | 10, 25, 26 | sylanbrc 695 | 1 ⊢ (𝜑 → 𝑉:𝐼⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 ↾s cress 15696 Ringcrg 18370 SubRingcsubrg 18599 mPwSer cmps 19172 mVar cmvr 19173 mPoly cmpl 19174 |
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-rmo 2904 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-int 4411 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-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 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-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-tset 15787 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-subg 17414 df-mgp 18313 df-ur 18325 df-ring 18372 df-subrg 18601 df-psr 19177 df-mvr 19178 df-mpl 19179 |
This theorem is referenced by: subrgvr1cl 19453 |
Copyright terms: Public domain | W3C validator |