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Mirrors > Home > MPE Home > Th. List > Mathboxes > selvval2lem5 | Structured version Visualization version GIF version |
Description: The fifth argument passed to evalSub is in the domain (a function 𝐼⟶𝐸). (Contributed by SN, 22-Feb-2024.) |
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) |
Ref | Expression |
---|---|
selvval2lem5 | ⊢ (𝜑 → 𝐹 ∈ (𝐸 ↑m 𝐼)) |
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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