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Mirrors > Home > MPE Home > Th. List > plydivlem2 | Structured version Visualization version GIF version |
Description: Lemma for plydivalg 24891. (Contributed by Mario Carneiro, 24-Jul-2014.) |
Ref | Expression |
---|---|
plydiv.pl | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
plydiv.tm | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) |
plydiv.rc | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆) |
plydiv.m1 | ⊢ (𝜑 → -1 ∈ 𝑆) |
plydiv.f | ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) |
plydiv.g | ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) |
plydiv.z | ⊢ (𝜑 → 𝐺 ≠ 0𝑝) |
plydiv.r | ⊢ 𝑅 = (𝐹 ∘f − (𝐺 ∘f · 𝑞)) |
Ref | Expression |
---|---|
plydivlem2 | ⊢ ((𝜑 ∧ 𝑞 ∈ (Poly‘𝑆)) → 𝑅 ∈ (Poly‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | plydiv.r | . 2 ⊢ 𝑅 = (𝐹 ∘f − (𝐺 ∘f · 𝑞)) | |
2 | plydiv.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) | |
3 | 2 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆)) |
4 | plydiv.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) | |
5 | 4 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ (Poly‘𝑆)) → 𝐺 ∈ (Poly‘𝑆)) |
6 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ (Poly‘𝑆)) → 𝑞 ∈ (Poly‘𝑆)) | |
7 | plydiv.pl | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
8 | 7 | adantlr 713 | . . . 4 ⊢ (((𝜑 ∧ 𝑞 ∈ (Poly‘𝑆)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
9 | plydiv.tm | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) | |
10 | 9 | adantlr 713 | . . . 4 ⊢ (((𝜑 ∧ 𝑞 ∈ (Poly‘𝑆)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) |
11 | 5, 6, 8, 10 | plymul 24811 | . . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ (Poly‘𝑆)) → (𝐺 ∘f · 𝑞) ∈ (Poly‘𝑆)) |
12 | plydiv.m1 | . . . 4 ⊢ (𝜑 → -1 ∈ 𝑆) | |
13 | 12 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ (Poly‘𝑆)) → -1 ∈ 𝑆) |
14 | 3, 11, 8, 10, 13 | plysub 24812 | . 2 ⊢ ((𝜑 ∧ 𝑞 ∈ (Poly‘𝑆)) → (𝐹 ∘f − (𝐺 ∘f · 𝑞)) ∈ (Poly‘𝑆)) |
15 | 1, 14 | eqeltrid 2920 | 1 ⊢ ((𝜑 ∧ 𝑞 ∈ (Poly‘𝑆)) → 𝑅 ∈ (Poly‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ‘cfv 6358 (class class class)co 7159 ∘f cof 7410 0cc0 10540 1c1 10541 + caddc 10543 · cmul 10545 − cmin 10873 -cneg 10874 / cdiv 11300 0𝑝c0p 24273 Polycply 24777 |
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 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-clim 14848 df-sum 15046 df-ply 24781 |
This theorem is referenced by: plydiveu 24890 |
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