Metamath Proof Explorer


Theorem selvcl

Description: Closure of the "variable selection" function. (Contributed by SN, 22-Feb-2024)

Ref Expression
Hypotheses selvcl.p P = I mPoly R
selvcl.b B = Base P
selvcl.u U = I J mPoly R
selvcl.t T = J mPoly U
selvcl.e E = Base T
selvcl.i φ I V
selvcl.r φ R CRing
selvcl.j φ J I
selvcl.f φ F B
Assertion selvcl φ I selectVars R J F E

Proof

Step Hyp Ref Expression
1 selvcl.p P = I mPoly R
2 selvcl.b B = Base P
3 selvcl.u U = I J mPoly R
4 selvcl.t T = J mPoly U
5 selvcl.e E = Base T
6 selvcl.i φ I V
7 selvcl.r φ R CRing
8 selvcl.j φ J I
9 selvcl.f φ F B
10 eqid algSc T = algSc T
11 eqid algSc T algSc U = algSc T algSc U
12 1 2 3 4 10 11 6 7 8 9 selvval φ I selectVars R J F = I evalSub T ran algSc T algSc U algSc T algSc U F x I if x J J mVar U x algSc T I J mVar R x
13 eqid T 𝑠 E I = T 𝑠 E I
14 eqid Base T 𝑠 E I = Base T 𝑠 E I
15 6 8 ssexd φ J V
16 6 difexd φ I J V
17 3 mplcrng I J V R CRing U CRing
18 16 7 17 syl2anc φ U CRing
19 4 mplcrng J V U CRing T CRing
20 15 18 19 syl2anc φ T CRing
21 ovexd φ E I V
22 eqid I evalSub T ran algSc T algSc U = I evalSub T ran algSc T algSc U
23 eqid I mPoly T 𝑠 ran algSc T algSc U = I mPoly T 𝑠 ran algSc T algSc U
24 eqid T 𝑠 ran algSc T algSc U = T 𝑠 ran algSc T algSc U
25 3 4 10 11 22 23 24 13 5 6 7 8 selvval2lemn φ I evalSub T ran algSc T algSc U I mPoly T 𝑠 ran algSc T algSc U RingHom T 𝑠 E I
26 eqid Base I mPoly T 𝑠 ran algSc T algSc U = Base I mPoly T 𝑠 ran algSc T algSc U
27 26 14 rhmf I evalSub T ran algSc T algSc U I mPoly T 𝑠 ran algSc T algSc U RingHom T 𝑠 E I I evalSub T ran algSc T algSc U : Base I mPoly T 𝑠 ran algSc T algSc U Base T 𝑠 E I
28 25 27 syl φ I evalSub T ran algSc T algSc U : Base I mPoly T 𝑠 ran algSc T algSc U Base T 𝑠 E I
29 1 2 3 4 10 11 24 23 26 6 7 8 9 selvval2lem4 φ algSc T algSc U F Base I mPoly T 𝑠 ran algSc T algSc U
30 28 29 ffvelrnd φ I evalSub T ran algSc T algSc U algSc T algSc U F Base T 𝑠 E I
31 13 5 14 20 21 30 pwselbas φ I evalSub T ran algSc T algSc U algSc T algSc U F : E I E
32 eqid x I if x J J mVar U x algSc T I J mVar R x = x I if x J J mVar U x algSc T I J mVar R x
33 3 4 10 5 32 6 7 8 selvval2lem5 φ x I if x J J mVar U x algSc T I J mVar R x E I
34 31 33 ffvelrnd φ I evalSub T ran algSc T algSc U algSc T algSc U F x I if x J J mVar U x algSc T I J mVar R x E
35 12 34 eqeltrd φ I selectVars R J F E