selvval2lem5 - Mathbox for Steven Nguyen

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Theorem selvval2lem5 40155

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 2738	. . . . . 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 5243	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ V)
7	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝐽 ∈ V)
8	4	difexd 5248	. . . . . . . 8 ⊢ (𝜑 → (𝐼 ∖ 𝐽) ∈ V)
9		selvval2lem5.r	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ CRing)
10		crngring 19710	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
11	9, 10	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
12		selvval2lem5.u	. . . . . . . . 9 ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)
13	12	mplring 21134	. . . . . . . 8 ⊢ (((𝐼 ∖ 𝐽) ∈ V ∧ 𝑅 ∈ Ring) → 𝑈 ∈ Ring)
14	8, 11, 13	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ Ring)
15	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑈 ∈ Ring)
16		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐽)
17	1, 2, 3, 7, 15, 16	mvrcl 21131	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝐽 mVar 𝑈)‘𝑥) ∈ 𝐸)
18	17	adantlr 711	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 ∈ 𝐽) → ((𝐽 mVar 𝑈)‘𝑥) ∈ 𝐸)
19		eqid 2738	. . . . . . 7 ⊢ (Base‘𝑈) = (Base‘𝑈)
20		selvval2lem5.c	. . . . . . 7 ⊢ 𝐶 = (algSc‘𝑇)
21	1, 3, 19, 20, 6, 14	mplasclf 21183	. . . . . 6 ⊢ (𝜑 → 𝐶:(Base‘𝑈)⟶𝐸)
22	21	ad2antrr 722	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → 𝐶:(Base‘𝑈)⟶𝐸)
23		eqid 2738	. . . . . 6 ⊢ ((𝐼 ∖ 𝐽) mVar 𝑅) = ((𝐼 ∖ 𝐽) mVar 𝑅)
24	8	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → (𝐼 ∖ 𝐽) ∈ V)
25	11	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → 𝑅 ∈ Ring)
26		eldif 3893	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐼 ∖ 𝐽) ↔ (𝑥 ∈ 𝐼 ∧ ¬ 𝑥 ∈ 𝐽))
27	26	biimpri 227	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐼 ∧ ¬ 𝑥 ∈ 𝐽) → 𝑥 ∈ (𝐼 ∖ 𝐽))
28	27	adantll 710	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → 𝑥 ∈ (𝐼 ∖ 𝐽))
29	12, 23, 19, 24, 25, 28	mvrcl 21131	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥) ∈ (Base‘𝑈))
30	22, 29	ffvelrnd 6944	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)) ∈ 𝐸)
31	18, 30	ifclda 4491	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))) ∈ 𝐸)
32		selvval2lem5.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))
33	31, 32	fmptd 6970	. 2 ⊢ (𝜑 → 𝐹:𝐼⟶𝐸)
34		fvexd 6771	. . . 4 ⊢ (𝜑 → (Base‘𝑇) ∈ V)
35	3, 34	eqeltrid 2843	. . 3 ⊢ (𝜑 → 𝐸 ∈ V)
36	35, 4	elmapd 8587	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝐸 ↑_m 𝐼) ↔ 𝐹:𝐼⟶𝐸))
37	33, 36	mpbird 256	1 ⊢ (𝜑 → 𝐹 ∈ (𝐸 ↑_m 𝐼))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 Vcvv 3422 ∖ cdif 3880 ⊆ wss 3883 ifcif 4456 ↦ cmpt 5153 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ↑_m cmap 8573 Basecbs 16840 Ringcrg 19698 CRingccrg 19699 algSccascl 20969 mVar cmvr 21018 mPoly cmpl 21019

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-ofr 7512 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-fzo 13312 df-seq 13650 df-hash 13973 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-tset 16907 df-0g 17069 df-gsum 17070 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-ghm 18747 df-cntz 18838 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-subrg 19937 df-lmod 20040 df-lss 20109 df-ascl 20972 df-psr 21022 df-mvr 21023 df-mpl 21024

This theorem is referenced by: selvcl 40156

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