selvcllem5 - Metamath Proof Explorer

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Theorem selvcllem5 22172

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

**Hypotheses**
Ref	Expression
selvcllem5.u	⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)
selvcllem5.t	⊢ 𝑇 = (𝐽 mPoly 𝑈)
selvcllem5.c	⊢ 𝐶 = (algSc‘𝑇)
selvcllem5.e	⊢ 𝐸 = (Base‘𝑇)
selvcllem5.f	⊢ 𝐹 = (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))
selvcllem5.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
selvcllem5.r	⊢ (𝜑 → 𝑅 ∈ CRing)
selvcllem5.j	⊢ (𝜑 → 𝐽 ⊆ 𝐼)

**Assertion**
Ref	Expression
selvcllem5	⊢ (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 selvcllem5**
Step	Hyp	Ref	Expression
1		selvcllem5.e	. . . 4 ⊢ 𝐸 = (Base‘𝑇)
2	1	fvexi 6877	. . 3 ⊢ 𝐸 ∈ V
3	2	a1i 11	. 2 ⊢ (𝜑 → 𝐸 ∈ V)
4		selvcllem5.i	. 2 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
5		selvcllem5.t	. . . . . 6 ⊢ 𝑇 = (𝐽 mPoly 𝑈)
6		eqid 2761	. . . . . 6 ⊢ (𝐽 mVar 𝑈) = (𝐽 mVar 𝑈)
7		selvcllem5.j	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ⊆ 𝐼)
8	4, 7	ssexd 5279	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ V)
9	8	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝐽 ∈ V)
10		selvcllem5.u	. . . . . . . 8 ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)
11	4	difexd 5286	. . . . . . . 8 ⊢ (𝜑 → (𝐼 ∖ 𝐽) ∈ V)
12		selvcllem5.r	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ CRing)
13	12	crngringd 20275	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
14	10, 11, 13	mplringd 22054	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ Ring)
15	14	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑈 ∈ Ring)
16		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐽)
17	5, 6, 1, 9, 15, 16	mvrcl 22023	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝐽 mVar 𝑈)‘𝑥) ∈ 𝐸)
18	17	adantlr 725	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 ∈ 𝐽) → ((𝐽 mVar 𝑈)‘𝑥) ∈ 𝐸)
19		eqid 2761	. . . . . . 7 ⊢ (Base‘𝑈) = (Base‘𝑈)
20		selvcllem5.c	. . . . . . 7 ⊢ 𝐶 = (algSc‘𝑇)
21	5, 1, 19, 20, 8, 14	mplasclf 22098	. . . . . 6 ⊢ (𝜑 → 𝐶:(Base‘𝑈)⟶𝐸)
22	21	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → 𝐶:(Base‘𝑈)⟶𝐸)
23		eqid 2761	. . . . . 6 ⊢ ((𝐼 ∖ 𝐽) mVar 𝑅) = ((𝐼 ∖ 𝐽) mVar 𝑅)
24	11	ad2antrr 736	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → (𝐼 ∖ 𝐽) ∈ V)
25	13	ad2antrr 736	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → 𝑅 ∈ Ring)
26		eldif 3914	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐼 ∖ 𝐽) ↔ (𝑥 ∈ 𝐼 ∧ ¬ 𝑥 ∈ 𝐽))
27	26	biimpri 230	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐼 ∧ ¬ 𝑥 ∈ 𝐽) → 𝑥 ∈ (𝐼 ∖ 𝐽))
28	27	adantll 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → 𝑥 ∈ (𝐼 ∖ 𝐽))
29	10, 23, 19, 24, 25, 28	mvrcl 22023	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥) ∈ (Base‘𝑈))
30	22, 29	ffvelcdmd 7062	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)) ∈ 𝐸)
31	18, 30	ifclda 4515	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))) ∈ 𝐸)
32		selvcllem5.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))
33	31, 32	fmptd 7091	. 2 ⊢ (𝜑 → 𝐹:𝐼⟶𝐸)
34	3, 4, 33	elmapdd 8818	1 ⊢ (𝜑 → 𝐹 ∈ (𝐸 ↑_m 𝐼))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 Vcvv 3453 ∖ cdif 3901 ⊆ wss 3904 ifcif 4479 ↦ cmpt 5180 ⟶wf 6513 ‘cfv 6517 (class class class)co 7392 ↑_m cmap 8803 Basecbs 17228 Ringcrg 20262 CRingccrg 20263 algSccascl 21884 mVar cmvr 21937 mPoly cmpl 21938

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-se 5599 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-isom 6526 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-of 7656 df-ofr 7657 df-om 7843 df-1st 7966 df-2nd 7967 df-supp 8136 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-er 8673 df-map 8805 df-pm 8806 df-ixp 8876 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-fsupp 9305 df-sup 9385 df-oi 9455 df-card 9894 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12479 df-z 12566 df-dec 12686 df-uz 12837 df-fz 13510 df-fzo 13657 df-seq 14012 df-hash 14341 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-mulr 17283 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-hom 17293 df-cco 17294 df-0g 17453 df-gsum 17454 df-prds 17459 df-pws 17461 df-mre 17597 df-mrc 17598 df-acs 17600 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-mhm 18800 df-submnd 18801 df-grp 18961 df-minusg 18962 df-sbg 18963 df-mulg 19093 df-subg 19148 df-ghm 19237 df-cntz 19340 df-cmn 19805 df-abl 19806 df-mgp 20170 df-rng 20182 df-ur 20211 df-ring 20264 df-cring 20265 df-subrng 20575 df-subrg 20599 df-lmod 20909 df-lss 20979 df-ascl 21887 df-psr 21941 df-mvr 21942 df-mpl 21943

This theorem is referenced by: selvcl 22173 selvadd 22176 selvmul 22177

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