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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 2777 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑆)) |
4 | 3 | psrbaspropd 22252 | . . . . 5 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑆))) |
5 | 4 | adantr 480 | . . . 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 18688 | . . . . . 6 ⊢ (𝜑 → (0g‘𝑅) = (0g‘𝑆)) |
8 | 7 | breq2d 5160 | . . . . 5 ⊢ (𝜑 → (𝑎 finSupp (0g‘𝑅) ↔ 𝑎 finSupp (0g‘𝑆))) |
9 | 8 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → (𝑎 finSupp (0g‘𝑅) ↔ 𝑎 finSupp (0g‘𝑆))) |
10 | 5, 9 | rabeqbidv 3452 | . . 3 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → {𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∣ 𝑎 finSupp (0g‘𝑅)} = {𝑎 ∈ (Base‘(𝐼 mPwSer 𝑆)) ∣ 𝑎 finSupp (0g‘𝑆)}) |
11 | eqid 2735 | . . . 4 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅) | |
12 | eqid 2735 | . . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
13 | eqid 2735 | . . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅)) | |
14 | eqid 2735 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
15 | eqid 2735 | . . . 4 ⊢ (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑅)) | |
16 | 11, 12, 13, 14, 15 | mplbas 22028 | . . 3 ⊢ (Base‘(𝐼 mPoly 𝑅)) = {𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∣ 𝑎 finSupp (0g‘𝑅)} |
17 | eqid 2735 | . . . 4 ⊢ (𝐼 mPoly 𝑆) = (𝐼 mPoly 𝑆) | |
18 | eqid 2735 | . . . 4 ⊢ (𝐼 mPwSer 𝑆) = (𝐼 mPwSer 𝑆) | |
19 | eqid 2735 | . . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑆)) = (Base‘(𝐼 mPwSer 𝑆)) | |
20 | eqid 2735 | . . . 4 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
21 | eqid 2735 | . . . 4 ⊢ (Base‘(𝐼 mPoly 𝑆)) = (Base‘(𝐼 mPoly 𝑆)) | |
22 | 17, 18, 19, 20, 21 | mplbas 22028 | . . 3 ⊢ (Base‘(𝐼 mPoly 𝑆)) = {𝑎 ∈ (Base‘(𝐼 mPwSer 𝑆)) ∣ 𝑎 finSupp (0g‘𝑆)} |
23 | 10, 16, 22 | 3eqtr4g 2800 | . 2 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑆))) |
24 | reldmmpl 22026 | . . . . . 6 ⊢ Rel dom mPoly | |
25 | 24 | ovprc1 7470 | . . . . 5 ⊢ (¬ 𝐼 ∈ V → (𝐼 mPoly 𝑅) = ∅) |
26 | 24 | ovprc1 7470 | . . . . 5 ⊢ (¬ 𝐼 ∈ V → (𝐼 mPoly 𝑆) = ∅) |
27 | 25, 26 | eqtr4d 2778 | . . . 4 ⊢ (¬ 𝐼 ∈ V → (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑆)) |
28 | 27 | fveq2d 6911 | . . 3 ⊢ (¬ 𝐼 ∈ V → (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑆))) |
29 | 28 | adantl 481 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑆))) |
30 | 23, 29 | pm2.61dan 813 | 1 ⊢ (𝜑 → (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {crab 3433 Vcvv 3478 ∅c0 4339 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 finSupp cfsupp 9399 Basecbs 17245 +gcplusg 17298 0gc0g 17486 mPwSer cmps 21942 mPoly cmpl 21944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-tset 17317 df-0g 17488 df-psr 21947 df-mpl 21949 |
This theorem is referenced by: ply1baspropd 22260 mdegpropd 26138 |
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