![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2773 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑆)) |
4 | 3 | psrbaspropd 21688 | . . . . 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 18563 | . . . . . 6 ⊢ (𝜑 → (0g‘𝑅) = (0g‘𝑆)) |
8 | 7 | breq2d 5153 | . . . . 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 3448 | . . 3 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → {𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∣ 𝑎 finSupp (0g‘𝑅)} = {𝑎 ∈ (Base‘(𝐼 mPwSer 𝑆)) ∣ 𝑎 finSupp (0g‘𝑆)}) |
11 | eqid 2731 | . . . 4 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅) | |
12 | eqid 2731 | . . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
13 | eqid 2731 | . . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅)) | |
14 | eqid 2731 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
15 | eqid 2731 | . . . 4 ⊢ (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑅)) | |
16 | 11, 12, 13, 14, 15 | mplbas 21480 | . . 3 ⊢ (Base‘(𝐼 mPoly 𝑅)) = {𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∣ 𝑎 finSupp (0g‘𝑅)} |
17 | eqid 2731 | . . . 4 ⊢ (𝐼 mPoly 𝑆) = (𝐼 mPoly 𝑆) | |
18 | eqid 2731 | . . . 4 ⊢ (𝐼 mPwSer 𝑆) = (𝐼 mPwSer 𝑆) | |
19 | eqid 2731 | . . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑆)) = (Base‘(𝐼 mPwSer 𝑆)) | |
20 | eqid 2731 | . . . 4 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
21 | eqid 2731 | . . . 4 ⊢ (Base‘(𝐼 mPoly 𝑆)) = (Base‘(𝐼 mPoly 𝑆)) | |
22 | 17, 18, 19, 20, 21 | mplbas 21480 | . . 3 ⊢ (Base‘(𝐼 mPoly 𝑆)) = {𝑎 ∈ (Base‘(𝐼 mPwSer 𝑆)) ∣ 𝑎 finSupp (0g‘𝑆)} |
23 | 10, 16, 22 | 3eqtr4g 2796 | . 2 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑆))) |
24 | reldmmpl 21478 | . . . . . 6 ⊢ Rel dom mPoly | |
25 | 24 | ovprc1 7432 | . . . . 5 ⊢ (¬ 𝐼 ∈ V → (𝐼 mPoly 𝑅) = ∅) |
26 | 24 | ovprc1 7432 | . . . . 5 ⊢ (¬ 𝐼 ∈ V → (𝐼 mPoly 𝑆) = ∅) |
27 | 25, 26 | eqtr4d 2774 | . . . 4 ⊢ (¬ 𝐼 ∈ V → (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑆)) |
28 | 27 | fveq2d 6882 | . . 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 811 | 1 ⊢ (𝜑 → (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {crab 3431 Vcvv 3473 ∅c0 4318 class class class wbr 5141 ‘cfv 6532 (class class class)co 7393 finSupp cfsupp 9344 Basecbs 17126 +gcplusg 17179 0gc0g 17367 mPwSer cmps 21388 mPoly cmpl 21390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-of 7653 df-om 7839 df-1st 7957 df-2nd 7958 df-supp 8129 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-map 8805 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-fsupp 9345 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-uz 12805 df-fz 13467 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-tset 17198 df-0g 17369 df-psr 21393 df-mpl 21395 |
This theorem is referenced by: ply1baspropd 21696 mdegpropd 25531 |
Copyright terms: Public domain | W3C validator |