eqrabdioph - Mathbox for Stefan O'Rear

	Mathbox for Stefan O'Rear		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > eqrabdioph		Structured version Visualization version GIF version

Theorem eqrabdioph 41505

Description: Diophantine set builder for equality of polynomial expressions. Note that the two expressions need not be nonnegative; only variables are so constrained. (Contributed by Stefan O'Rear, 10-Oct-2014.)

**Assertion**
Ref	Expression
eqrabdioph	⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ 𝐴 = 𝐵} ∈ (Dioph‘𝑁))

Distinct variable group: 𝑡,𝑁 Allowed substitution hints: 𝐴(𝑡) 𝐵(𝑡)

**Proof of Theorem eqrabdioph**
Step	Hyp	Ref	Expression
1		nfmpt1 5256	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴)
2	1	nfel1 2919	. . . . . 6 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))
3		nfmpt1 5256	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵)
4	3	nfel1 2919	. . . . . 6 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))
5	2, 4	nfan 1902	. . . . 5 ⊢ Ⅎ𝑡((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁)))
6		mzpf 41464	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴):(ℤ ↑_m (1...𝑁))⟶ℤ)
7	6	ad2antrr 724	. . . . . . . . . 10 ⊢ ((((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))) → (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴):(ℤ ↑_m (1...𝑁))⟶ℤ)
8		zex 12566	. . . . . . . . . . . . 13 ⊢ ℤ ∈ V
9		nn0ssz 12580	. . . . . . . . . . . . 13 ⊢ ℕ₀ ⊆ ℤ
10		mapss 8882	. . . . . . . . . . . . 13 ⊢ ((ℤ ∈ V ∧ ℕ₀ ⊆ ℤ) → (ℕ₀ ↑_m (1...𝑁)) ⊆ (ℤ ↑_m (1...𝑁)))
11	8, 9, 10	mp2an 690	. . . . . . . . . . . 12 ⊢ (ℕ₀ ↑_m (1...𝑁)) ⊆ (ℤ ↑_m (1...𝑁))
12	11	sseli 3978	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) → 𝑡 ∈ (ℤ ↑_m (1...𝑁)))
13	12	adantl 482	. . . . . . . . . 10 ⊢ ((((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))) → 𝑡 ∈ (ℤ ↑_m (1...𝑁)))
14		mptfcl 41448	. . . . . . . . . 10 ⊢ ((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴):(ℤ ↑_m (1...𝑁))⟶ℤ → (𝑡 ∈ (ℤ ↑_m (1...𝑁)) → 𝐴 ∈ ℤ))
15	7, 13, 14	sylc 65	. . . . . . . . 9 ⊢ ((((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))) → 𝐴 ∈ ℤ)
16	15	zcnd 12666	. . . . . . . 8 ⊢ ((((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))) → 𝐴 ∈ ℂ)
17		mzpf 41464	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁)) → (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵):(ℤ ↑_m (1...𝑁))⟶ℤ)
18	17	ad2antlr 725	. . . . . . . . . 10 ⊢ ((((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))) → (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵):(ℤ ↑_m (1...𝑁))⟶ℤ)
19		mptfcl 41448	. . . . . . . . . 10 ⊢ ((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵):(ℤ ↑_m (1...𝑁))⟶ℤ → (𝑡 ∈ (ℤ ↑_m (1...𝑁)) → 𝐵 ∈ ℤ))
20	18, 13, 19	sylc 65	. . . . . . . . 9 ⊢ ((((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))) → 𝐵 ∈ ℤ)
21	20	zcnd 12666	. . . . . . . 8 ⊢ ((((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))) → 𝐵 ∈ ℂ)
22	16, 21	subeq0ad 11580	. . . . . . 7 ⊢ ((((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵))
23	22	bicomd 222	. . . . . 6 ⊢ ((((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))) → (𝐴 = 𝐵 ↔ (𝐴 − 𝐵) = 0))
24	23	ex 413	. . . . 5 ⊢ (((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) → (𝐴 = 𝐵 ↔ (𝐴 − 𝐵) = 0)))
25	5, 24	ralrimi 3254	. . . 4 ⊢ (((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → ∀𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))(𝐴 = 𝐵 ↔ (𝐴 − 𝐵) = 0))
26		rabbi 3462	. . . 4 ⊢ (∀𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))(𝐴 = 𝐵 ↔ (𝐴 − 𝐵) = 0) ↔ {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ 𝐴 = 𝐵} = {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ (𝐴 − 𝐵) = 0})
27	25, 26	sylib 217	. . 3 ⊢ (((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ 𝐴 = 𝐵} = {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ (𝐴 − 𝐵) = 0})
28	27	3adant1 1130	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ 𝐴 = 𝐵} = {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ (𝐴 − 𝐵) = 0})
29		simp1 1136	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → 𝑁 ∈ ℕ₀)
30		mzpsubmpt 41471	. . . 4 ⊢ (((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ (𝐴 − 𝐵)) ∈ (mzPoly‘(1...𝑁)))
31	30	3adant1 1130	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ (𝐴 − 𝐵)) ∈ (mzPoly‘(1...𝑁)))
32		eq0rabdioph 41504	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ (𝐴 − 𝐵)) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ (𝐴 − 𝐵) = 0} ∈ (Dioph‘𝑁))
33	29, 31, 32	syl2anc 584	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ (𝐴 − 𝐵) = 0} ∈ (Dioph‘𝑁))
34	28, 33	eqeltrd 2833	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ 𝐴 = 𝐵} ∈ (Dioph‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 {crab 3432 Vcvv 3474 ⊆ wss 3948 ↦ cmpt 5231 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 ↑_m cmap 8819 0cc0 11109 1c1 11110 − cmin 11443 ℕ₀cn0 12471 ℤcz 12557 ...cfz 13483 mzPolycmzp 41450 Diophcdioph 41483

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13484 df-mzpcl 41451 df-mzp 41452 df-dioph 41484

This theorem is referenced by: elnn0rabdioph 41531 dvdsrabdioph 41538 rmydioph 41743 rmxdioph 41745 expdiophlem2 41751 expdioph 41752

W3C validator