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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 4090 | . . 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 2735 | . . . . 5 ⊢ {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } | |
8 | 3, 4, 5, 6, 7 | mplval 22027 | . . . 4 ⊢ 𝑃 = (𝑆 ↾s {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 }) |
9 | 8, 5 | ressbas2 17283 | . . 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 2766 | 1 ⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp 0 } |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 {crab 3433 ⊆ wss 3963 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 finSupp cfsupp 9399 Basecbs 17245 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-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-1cn 11211 ax-addcl 11213 |
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-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-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-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-nn 12265 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-psr 21947 df-mpl 21949 |
This theorem is referenced by: mplelbas 22029 mplval2 22034 mplbasss 22035 mplsubglem2 22039 ressmplbas2 22063 mplbaspropd 22254 |
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