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Mirrors > Home > MPE Home > Th. List > ply0 | Structured version Visualization version GIF version |
Description: The zero function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.) |
Ref | Expression |
---|---|
ply0 | ⊢ (𝑆 ⊆ ℂ → 0𝑝 ∈ (Poly‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-0p 25419 | . . 3 ⊢ 0𝑝 = (ℂ × {0}) | |
2 | id 22 | . . . . 5 ⊢ (𝑆 ⊆ ℂ → 𝑆 ⊆ ℂ) | |
3 | 0cnd 11211 | . . . . . 6 ⊢ (𝑆 ⊆ ℂ → 0 ∈ ℂ) | |
4 | 3 | snssd 4811 | . . . . 5 ⊢ (𝑆 ⊆ ℂ → {0} ⊆ ℂ) |
5 | 2, 4 | unssd 4185 | . . . 4 ⊢ (𝑆 ⊆ ℂ → (𝑆 ∪ {0}) ⊆ ℂ) |
6 | ssun2 4172 | . . . . 5 ⊢ {0} ⊆ (𝑆 ∪ {0}) | |
7 | c0ex 11212 | . . . . . 6 ⊢ 0 ∈ V | |
8 | 7 | snss 4788 | . . . . 5 ⊢ (0 ∈ (𝑆 ∪ {0}) ↔ {0} ⊆ (𝑆 ∪ {0})) |
9 | 6, 8 | mpbir 230 | . . . 4 ⊢ 0 ∈ (𝑆 ∪ {0}) |
10 | plyconst 25955 | . . . 4 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ 0 ∈ (𝑆 ∪ {0})) → (ℂ × {0}) ∈ (Poly‘(𝑆 ∪ {0}))) | |
11 | 5, 9, 10 | sylancl 584 | . . 3 ⊢ (𝑆 ⊆ ℂ → (ℂ × {0}) ∈ (Poly‘(𝑆 ∪ {0}))) |
12 | 1, 11 | eqeltrid 2835 | . 2 ⊢ (𝑆 ⊆ ℂ → 0𝑝 ∈ (Poly‘(𝑆 ∪ {0}))) |
13 | plyun0 25946 | . 2 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆) | |
14 | 12, 13 | eleqtrdi 2841 | 1 ⊢ (𝑆 ⊆ ℂ → 0𝑝 ∈ (Poly‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2104 ∪ cun 3945 ⊆ wss 3947 {csn 4627 × cxp 5673 ‘cfv 6542 ℂcc 11110 0cc0 11112 0𝑝c0p 25418 Polycply 25933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12979 df-fz 13489 df-fzo 13632 df-seq 13971 df-exp 14032 df-hash 14295 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-sum 15637 df-0p 25419 df-ply 25937 |
This theorem is referenced by: coe0 26005 plydivlem3 26044 |
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