Metamath Proof Explorer


Theorem selvval2lem5

Description: The fifth argument passed to evalSub is in the domain (a function I --> E ). (Contributed by SN, 22-Feb-2024)

Ref Expression
Hypotheses selvval2lem5.u U = I J mPoly R
selvval2lem5.t T = J mPoly U
selvval2lem5.c C = algSc T
selvval2lem5.e E = Base T
selvval2lem5.f F = x I if x J J mVar U x C I J mVar R x
selvval2lem5.i φ I V
selvval2lem5.r φ R CRing
selvval2lem5.j φ J I
Assertion selvval2lem5 φ F E I

Proof

Step Hyp Ref Expression
1 selvval2lem5.u U = I J mPoly R
2 selvval2lem5.t T = J mPoly U
3 selvval2lem5.c C = algSc T
4 selvval2lem5.e E = Base T
5 selvval2lem5.f F = x I if x J J mVar U x C I J mVar R x
6 selvval2lem5.i φ I V
7 selvval2lem5.r φ R CRing
8 selvval2lem5.j φ J I
9 eqid J mVar U = J mVar U
10 6 8 ssexd φ J V
11 10 adantr φ x J J V
12 6 difexd φ I J V
13 crngring R CRing R Ring
14 7 13 syl φ R Ring
15 1 mplring I J V R Ring U Ring
16 12 14 15 syl2anc φ U Ring
17 16 adantr φ x J U Ring
18 simpr φ x J x J
19 2 9 4 11 17 18 mvrcl φ x J J mVar U x E
20 19 adantlr φ x I x J J mVar U x E
21 eqid Base U = Base U
22 2 4 21 3 10 16 mplasclf φ C : Base U E
23 22 ad2antrr φ x I ¬ x J C : Base U E
24 eqid I J mVar R = I J mVar R
25 12 ad2antrr φ x I ¬ x J I J V
26 14 ad2antrr φ x I ¬ x J R Ring
27 eldif x I J x I ¬ x J
28 27 biimpri x I ¬ x J x I J
29 28 adantll φ x I ¬ x J x I J
30 1 24 21 25 26 29 mvrcl φ x I ¬ x J I J mVar R x Base U
31 23 30 ffvelrnd φ x I ¬ x J C I J mVar R x E
32 20 31 ifclda φ x I if x J J mVar U x C I J mVar R x E
33 32 5 fmptd φ F : I E
34 fvexd φ Base T V
35 4 34 eqeltrid φ E V
36 35 6 elmapd φ F E I F : I E
37 33 36 mpbird φ F E I