selvval2lem5 - Mathbox for Steven Nguyen

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Theorem selvval2lem5 39186

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

**Hypotheses**
Ref	Expression
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**
Ref	Expression
selvval2lem5	⊢ (𝜑 → 𝐹 ∈ (𝐸 ↑_m 𝐼))

Distinct variable groups: 𝑥,𝐸 𝑥,𝐼 𝜑,𝑥 Allowed substitution hints: 𝐶(𝑥) 𝑅(𝑥) 𝑇(𝑥) 𝑈(𝑥) 𝐹(𝑥) 𝐽(𝑥) 𝑉(𝑥)

**Proof of Theorem selvval2lem5**
Step	Hyp	Ref	Expression
1		selvval2lem5.t	. . . . . 6 ⊢ 𝑇 = (𝐽 mPoly 𝑈)
2		eqid 2821	. . . . . 6 ⊢ (𝐽 mVar 𝑈) = (𝐽 mVar 𝑈)
3		selvval2lem5.e	. . . . . 6 ⊢ 𝐸 = (Base‘𝑇)
4		selvval2lem5.i	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
5		selvval2lem5.j	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ⊆ 𝐼)
6	4, 5	ssexd 5228	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ V)
7	6	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝐽 ∈ V)
8		difexg 5231	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝑉 → (𝐼 ∖ 𝐽) ∈ V)
9	4, 8	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐼 ∖ 𝐽) ∈ V)
10		selvval2lem5.r	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ CRing)
11		crngring 19308	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
12	10, 11	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
13		selvval2lem5.u	. . . . . . . . 9 ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)
14	13	mplring 20232	. . . . . . . 8 ⊢ (((𝐼 ∖ 𝐽) ∈ V ∧ 𝑅 ∈ Ring) → 𝑈 ∈ Ring)
15	9, 12, 14	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ Ring)
16	15	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑈 ∈ Ring)
17		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐽)
18	1, 2, 3, 7, 16, 17	mvrcl 20229	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝐽 mVar 𝑈)‘𝑥) ∈ 𝐸)
19	18	adantlr 713	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 ∈ 𝐽) → ((𝐽 mVar 𝑈)‘𝑥) ∈ 𝐸)
20		eqid 2821	. . . . . . 7 ⊢ (Base‘𝑈) = (Base‘𝑈)
21		selvval2lem5.c	. . . . . . 7 ⊢ 𝐶 = (algSc‘𝑇)
22	1, 3, 20, 21, 6, 15	mplasclf 20277	. . . . . 6 ⊢ (𝜑 → 𝐶:(Base‘𝑈)⟶𝐸)
23	22	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → 𝐶:(Base‘𝑈)⟶𝐸)
24		eqid 2821	. . . . . 6 ⊢ ((𝐼 ∖ 𝐽) mVar 𝑅) = ((𝐼 ∖ 𝐽) mVar 𝑅)
25	9	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → (𝐼 ∖ 𝐽) ∈ V)
26	12	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → 𝑅 ∈ Ring)
27		eldif 3946	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐼 ∖ 𝐽) ↔ (𝑥 ∈ 𝐼 ∧ ¬ 𝑥 ∈ 𝐽))
28	27	biimpri 230	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐼 ∧ ¬ 𝑥 ∈ 𝐽) → 𝑥 ∈ (𝐼 ∖ 𝐽))
29	28	adantll 712	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → 𝑥 ∈ (𝐼 ∖ 𝐽))
30	13, 24, 20, 25, 26, 29	mvrcl 20229	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥) ∈ (Base‘𝑈))
31	23, 30	ffvelrnd 6852	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)) ∈ 𝐸)
32	19, 31	ifclda 4501	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))) ∈ 𝐸)
33		selvval2lem5.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))
34	32, 33	fmptd 6878	. 2 ⊢ (𝜑 → 𝐹:𝐼⟶𝐸)
35		fvexd 6685	. . . 4 ⊢ (𝜑 → (Base‘𝑇) ∈ V)
36	3, 35	eqeltrid 2917	. . 3 ⊢ (𝜑 → 𝐸 ∈ V)
37	36, 4	elmapd 8420	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝐸 ↑_m 𝐼) ↔ 𝐹:𝐼⟶𝐸))
38	34, 37	mpbird 259	1 ⊢ (𝜑 → 𝐹 ∈ (𝐸 ↑_m 𝐼))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∖ cdif 3933 ⊆ wss 3936 ifcif 4467 ↦ cmpt 5146 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ↑_m cmap 8406 Basecbs 16483 Ringcrg 19297 CRingccrg 19298 algSccascl 20084 mVar cmvr 20132 mPoly cmpl 20133

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-ofr 7410 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-tset 16584 df-0g 16715 df-gsum 16716 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mulg 18225 df-subg 18276 df-ghm 18356 df-cntz 18447 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-subrg 19533 df-lmod 19636 df-lss 19704 df-ascl 20087 df-psr 20136 df-mvr 20137 df-mpl 20138

This theorem is referenced by: selvcl 39187

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