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Mirrors > Home > MPE Home > Th. List > mplbas | Structured version Visualization version GIF version |
Description: Base set of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 25-Jun-2019.) |
Ref | Expression |
---|---|
mplval.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mplval.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
mplval.b | ⊢ 𝐵 = (Base‘𝑆) |
mplval.z | ⊢ 0 = (0g‘𝑅) |
mplbas.u | ⊢ 𝑈 = (Base‘𝑃) |
Ref | Expression |
---|---|
mplbas | ⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mplbas.u | . 2 ⊢ 𝑈 = (Base‘𝑃) | |
2 | ssrab2 3687 | . . 3 ⊢ {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } ⊆ 𝐵 | |
3 | mplval.p | . . . . 5 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
4 | mplval.s | . . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
5 | mplval.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑆) | |
6 | mplval.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
7 | eqid 2622 | . . . . 5 ⊢ {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } | |
8 | 3, 4, 5, 6, 7 | mplval 19428 | . . . 4 ⊢ 𝑃 = (𝑆 ↾s {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 }) |
9 | 8, 5 | ressbas2 15931 | . . 3 ⊢ ({𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } ⊆ 𝐵 → {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } = (Base‘𝑃)) |
10 | 2, 9 | ax-mp 5 | . 2 ⊢ {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } = (Base‘𝑃) |
11 | 1, 10 | eqtr4i 2647 | 1 ⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 {crab 2916 ⊆ wss 3574 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 finSupp cfsupp 8275 Basecbs 15857 0gc0g 16100 mPwSer cmps 19351 mPoly cmpl 19353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-i2m1 10004 ax-1ne0 10005 ax-rrecex 10008 ax-cnre 10009 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-nn 11021 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-psr 19356 df-mpl 19358 |
This theorem is referenced by: mplelbas 19430 mplval2 19431 mplbasss 19432 mplsubglem2 19436 ressmplbas2 19455 mplbaspropd 19607 |
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