selvcllem5 - Metamath Proof Explorer

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Theorem selvcllem5 22453

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 6899	. . 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 2760	. . . . . 6 ⊢ (𝐽 mVar 𝑈) = (𝐽 mVar 𝑈)
7		selvcllem5.j	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ⊆ 𝐼)
8	4, 7	ssexd 5285	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ V)
9	8	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝐽 ∈ V)
10		selvcllem5.u	. . . . . . . 8 ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)
11	4	difexd 5292	. . . . . . . 8 ⊢ (𝜑 → (𝐼 ∖ 𝐽) ∈ V)
12		selvcllem5.r	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ CRing)
13	12	crngringd 20478	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
14	10, 11, 13	mplringd 22335	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ Ring)
15	14	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑈 ∈ Ring)
16		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐽)
17	5, 6, 1, 9, 15, 16	mvrcl 22304	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝐽 mVar 𝑈)‘𝑥) ∈ 𝐸)
18	17	adantlr 728	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 ∈ 𝐽) → ((𝐽 mVar 𝑈)‘𝑥) ∈ 𝐸)
19		eqid 2760	. . . . . . 7 ⊢ (Base‘𝑈) = (Base‘𝑈)
20		selvcllem5.c	. . . . . . 7 ⊢ 𝐶 = (algSc‘𝑇)
21	5, 1, 19, 20, 8, 14	mplasclf 22379	. . . . . 6 ⊢ (𝜑 → 𝐶:(Base‘𝑈)⟶𝐸)
22	21	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → 𝐶:(Base‘𝑈)⟶𝐸)
23		eqid 2760	. . . . . 6 ⊢ ((𝐼 ∖ 𝐽) mVar 𝑅) = ((𝐼 ∖ 𝐽) mVar 𝑅)
24	11	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → (𝐼 ∖ 𝐽) ∈ V)
25	13	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → 𝑅 ∈ Ring)
26		eldif 4049	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐼 ∖ 𝐽) ↔ (𝑥 ∈ 𝐼 ∧ ¬ 𝑥 ∈ 𝐽))
27	26	biimpri 231	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐼 ∧ ¬ 𝑥 ∈ 𝐽) → 𝑥 ∈ (𝐼 ∖ 𝐽))
28	27	adantll 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → 𝑥 ∈ (𝐼 ∖ 𝐽))
29	10, 23, 19, 24, 25, 28	mvrcl 22304	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥) ∈ (Base‘𝑈))
30	22, 29	ffvelcdmd 7085	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)) ∈ 𝐸)
31	18, 30	ifclda 4517	. . 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 7114	. 2 ⊢ (𝜑 → 𝐹:𝐼⟶𝐸)
34	3, 4, 33	elmapdd 8864	1 ⊢ (𝜑 → 𝐹 ∈ (𝐸 ↑_m 𝐼))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 Vcvv 3450 ⊆ wss 3895 ∖ cdif 4045 ifcif 4481 ↦ cmpt 5185 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ↑_m cmap 8850 Basecbs 17392 Ringcrg 20464 CRingccrg 20465 algSccascl 22165 mVar cmvr 22218 mPoly cmpl 22219

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-ofr 7694 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8180 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-1o 8479 df-2o 8480 df-er 8720 df-map 8852 df-pm 8853 df-ixp 8929 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-fsupp 9357 df-sup 9437 df-oi 9507 df-card 10023 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-nn 12339 df-2 12408 df-3 12409 df-4 12410 df-5 12411 df-6 12412 df-7 12413 df-8 12414 df-9 12415 df-n0 12610 df-z 12697 df-dec 12818 df-uz 12969 df-fz 13643 df-fzo 13792 df-seq 14148 df-hash 14478 df-struct 17330 df-sets 17347 df-slot 17365 df-ndx 17377 df-base 17393 df-ress 17414 df-plusg 17446 df-mulr 17447 df-sca 17449 df-vsca 17450 df-ip 17451 df-tset 17452 df-ple 17453 df-ds 17455 df-hom 17457 df-cco 17458 df-0g 17617 df-gsum 17618 df-prds 17623 df-pws 17625 df-mre 17761 df-mrc 17762 df-acs 17764 df-mgm 18821 df-sgrp 18913 df-mnd 18929 df-mhm 18983 df-submnd 18984 df-grp 19152 df-minusg 19153 df-sbg 19154 df-mulg 19283 df-subg 19338 df-ghm 19433 df-cntz 19536 df-cmn 20001 df-abl 20002 df-mgp 20366 df-rng 20380 df-ur 20413 df-ring 20466 df-cring 20467 df-subrng 20803 df-subrg 20827 df-lmod 21142 df-lss 21212 df-ascl 22168 df-psr 22222 df-mvr 22223 df-mpl 22224

This theorem is used by: selvcl 22454 selvadd 22457 selvmul 22458

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