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Mirrors > Home > MPE Home > Th. List > mplbaspropd | Structured version Visualization version GIF version |
Description: Property deduction for polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 19-Jul-2019.) |
Ref | Expression |
---|---|
psrplusgpropd.b1 | ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) |
psrplusgpropd.b2 | ⊢ (𝜑 → 𝐵 = (Base‘𝑆)) |
psrplusgpropd.p | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑆)𝑦)) |
Ref | Expression |
---|---|
mplbaspropd | ⊢ (𝜑 → (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrplusgpropd.b1 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) | |
2 | psrplusgpropd.b2 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝑆)) | |
3 | 1, 2 | eqtr3d 2855 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑆)) |
4 | 3 | psrbaspropd 20331 | . . . . 5 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑆))) |
5 | 4 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑆))) |
6 | psrplusgpropd.p | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑆)𝑦)) | |
7 | 1, 2, 6 | grpidpropd 17860 | . . . . . 6 ⊢ (𝜑 → (0g‘𝑅) = (0g‘𝑆)) |
8 | 7 | breq2d 5069 | . . . . 5 ⊢ (𝜑 → (𝑎 finSupp (0g‘𝑅) ↔ 𝑎 finSupp (0g‘𝑆))) |
9 | 8 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → (𝑎 finSupp (0g‘𝑅) ↔ 𝑎 finSupp (0g‘𝑆))) |
10 | 5, 9 | rabeqbidv 3483 | . . 3 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → {𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∣ 𝑎 finSupp (0g‘𝑅)} = {𝑎 ∈ (Base‘(𝐼 mPwSer 𝑆)) ∣ 𝑎 finSupp (0g‘𝑆)}) |
11 | eqid 2818 | . . . 4 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅) | |
12 | eqid 2818 | . . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
13 | eqid 2818 | . . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅)) | |
14 | eqid 2818 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
15 | eqid 2818 | . . . 4 ⊢ (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑅)) | |
16 | 11, 12, 13, 14, 15 | mplbas 20137 | . . 3 ⊢ (Base‘(𝐼 mPoly 𝑅)) = {𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∣ 𝑎 finSupp (0g‘𝑅)} |
17 | eqid 2818 | . . . 4 ⊢ (𝐼 mPoly 𝑆) = (𝐼 mPoly 𝑆) | |
18 | eqid 2818 | . . . 4 ⊢ (𝐼 mPwSer 𝑆) = (𝐼 mPwSer 𝑆) | |
19 | eqid 2818 | . . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑆)) = (Base‘(𝐼 mPwSer 𝑆)) | |
20 | eqid 2818 | . . . 4 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
21 | eqid 2818 | . . . 4 ⊢ (Base‘(𝐼 mPoly 𝑆)) = (Base‘(𝐼 mPoly 𝑆)) | |
22 | 17, 18, 19, 20, 21 | mplbas 20137 | . . 3 ⊢ (Base‘(𝐼 mPoly 𝑆)) = {𝑎 ∈ (Base‘(𝐼 mPwSer 𝑆)) ∣ 𝑎 finSupp (0g‘𝑆)} |
23 | 10, 16, 22 | 3eqtr4g 2878 | . 2 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑆))) |
24 | reldmmpl 20135 | . . . . . 6 ⊢ Rel dom mPoly | |
25 | 24 | ovprc1 7184 | . . . . 5 ⊢ (¬ 𝐼 ∈ V → (𝐼 mPoly 𝑅) = ∅) |
26 | 24 | ovprc1 7184 | . . . . 5 ⊢ (¬ 𝐼 ∈ V → (𝐼 mPoly 𝑆) = ∅) |
27 | 25, 26 | eqtr4d 2856 | . . . 4 ⊢ (¬ 𝐼 ∈ V → (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑆)) |
28 | 27 | fveq2d 6667 | . . 3 ⊢ (¬ 𝐼 ∈ V → (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑆))) |
29 | 28 | adantl 482 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑆))) |
30 | 23, 29 | pm2.61dan 809 | 1 ⊢ (𝜑 → (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {crab 3139 Vcvv 3492 ∅c0 4288 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 finSupp cfsupp 8821 Basecbs 16471 +gcplusg 16553 0gc0g 16701 mPwSer cmps 20059 mPoly cmpl 20061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-tset 16572 df-0g 16703 df-psr 20064 df-mpl 20066 |
This theorem is referenced by: ply1baspropd 20339 mdegpropd 24605 |
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