eqrabdioph - Mathbox for Stefan O'Rear

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Theorem eqrabdioph 42098

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 5249	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴)
2	1	nfel1 2913	. . . . . 6 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))
3		nfmpt1 5249	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵)
4	3	nfel1 2913	. . . . . 6 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))
5	2, 4	nfan 1894	. . . . 5 ⊢ Ⅎ𝑡((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁)))
6		mzpf 42057	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴):(ℤ ↑_m (1...𝑁))⟶ℤ)
7	6	ad2antrr 723	. . . . . . . . . 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 12571	. . . . . . . . . . . . 13 ⊢ ℤ ∈ V
9		nn0ssz 12585	. . . . . . . . . . . . 13 ⊢ ℕ₀ ⊆ ℤ
10		mapss 8885	. . . . . . . . . . . . 13 ⊢ ((ℤ ∈ V ∧ ℕ₀ ⊆ ℤ) → (ℕ₀ ↑_m (1...𝑁)) ⊆ (ℤ ↑_m (1...𝑁)))
11	8, 9, 10	mp2an 689	. . . . . . . . . . . 12 ⊢ (ℕ₀ ↑_m (1...𝑁)) ⊆ (ℤ ↑_m (1...𝑁))
12	11	sseli 3973	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) → 𝑡 ∈ (ℤ ↑_m (1...𝑁)))
13	12	adantl 481	. . . . . . . . . 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 42041	. . . . . . . . . 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 12671	. . . . . . . 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 42057	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁)) → (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐵):(ℤ ↑_m (1...𝑁))⟶ℤ)
18	17	ad2antlr 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)) ↦ 𝐵):(ℤ ↑_m (1...𝑁))⟶ℤ)
19		mptfcl 42041	. . . . . . . . . 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 12671	. . . . . . . 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 11585	. . . . . . 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 412	. . . . 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 3248	. . . 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 3456	. . . 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 1127	. 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 1133	. . 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 42064	. . . 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 1127	. . 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 42097	. . 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 583	. 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 2827	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 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3055 {crab 3426 Vcvv 3468 ⊆ wss 3943 ↦ cmpt 5224 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 ↑_m cmap 8822 0cc0 11112 1c1 11113 − cmin 11448 ℕ₀cn0 12476 ℤcz 12562 ...cfz 13490 mzPolycmzp 42043 Diophcdioph 42076

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 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

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-mzpcl 42044 df-mzp 42045 df-dioph 42077

This theorem is referenced by: elnn0rabdioph 42124 dvdsrabdioph 42131 rmydioph 42336 rmxdioph 42338 expdiophlem2 42344 expdioph 42345

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