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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 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 5192 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ V) |
7 | 6 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝐽 ∈ V) |
8 | 4 | difexd 5197 | . . . . . . . 8 ⊢ (𝜑 → (𝐼 ∖ 𝐽) ∈ V) |
9 | selvval2lem5.r | . . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
10 | crngring 19428 | . . . . . . . . 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 20834 | . . . . . . . 8 ⊢ (((𝐼 ∖ 𝐽) ∈ V ∧ 𝑅 ∈ Ring) → 𝑈 ∈ Ring) |
14 | 8, 11, 13 | syl2anc 587 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ Ring) |
15 | 14 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑈 ∈ Ring) |
16 | simpr 488 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐽) | |
17 | 1, 2, 3, 7, 15, 16 | mvrcl 20831 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝐽 mVar 𝑈)‘𝑥) ∈ 𝐸) |
18 | 17 | adantlr 715 | . . . 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 20877 | . . . . . 6 ⊢ (𝜑 → 𝐶:(Base‘𝑈)⟶𝐸) |
22 | 21 | ad2antrr 726 | . . . . 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 726 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → (𝐼 ∖ 𝐽) ∈ V) |
25 | 11 | ad2antrr 726 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → 𝑅 ∈ Ring) |
26 | eldif 3853 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝐼 ∖ 𝐽) ↔ (𝑥 ∈ 𝐼 ∧ ¬ 𝑥 ∈ 𝐽)) | |
27 | 26 | biimpri 231 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐼 ∧ ¬ 𝑥 ∈ 𝐽) → 𝑥 ∈ (𝐼 ∖ 𝐽)) |
28 | 27 | adantll 714 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → 𝑥 ∈ (𝐼 ∖ 𝐽)) |
29 | 12, 23, 19, 24, 25, 28 | mvrcl 20831 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → (((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥) ∈ (Base‘𝑈)) |
30 | 22, 29 | ffvelrnd 6862 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 ∈ 𝐽) → (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)) ∈ 𝐸) |
31 | 18, 30 | ifclda 4449 | . . 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 6888 | . 2 ⊢ (𝜑 → 𝐹:𝐼⟶𝐸) |
34 | fvexd 6689 | . . . 4 ⊢ (𝜑 → (Base‘𝑇) ∈ V) | |
35 | 3, 34 | eqeltrid 2837 | . . 3 ⊢ (𝜑 → 𝐸 ∈ V) |
36 | 35, 4 | elmapd 8451 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝐸 ↑m 𝐼) ↔ 𝐹:𝐼⟶𝐸)) |
37 | 33, 36 | mpbird 260 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝐸 ↑m 𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 Vcvv 3398 ∖ cdif 3840 ⊆ wss 3843 ifcif 4414 ↦ cmpt 5110 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 ↑m cmap 8437 Basecbs 16586 Ringcrg 19416 CRingccrg 19417 algSccascl 20668 mVar cmvr 20718 mPoly cmpl 20719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-ofr 7426 df-om 7600 df-1st 7714 df-2nd 7715 df-supp 7857 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-map 8439 df-pm 8440 df-ixp 8508 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-fsupp 8907 df-oi 9047 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-uz 12325 df-fz 12982 df-fzo 13125 df-seq 13461 df-hash 13783 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-sca 16684 df-vsca 16685 df-tset 16687 df-0g 16818 df-gsum 16819 df-mre 16960 df-mrc 16961 df-acs 16963 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-mhm 18072 df-submnd 18073 df-grp 18222 df-minusg 18223 df-sbg 18224 df-mulg 18343 df-subg 18394 df-ghm 18474 df-cntz 18565 df-cmn 19026 df-abl 19027 df-mgp 19359 df-ur 19371 df-ring 19418 df-cring 19419 df-subrg 19652 df-lmod 19755 df-lss 19823 df-ascl 20671 df-psr 20722 df-mvr 20723 df-mpl 20724 |
This theorem is referenced by: selvcl 39812 |
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