mzpindd - Mathbox for Stefan O'Rear

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Theorem mzpindd 41786

Description: "Structural" induction to prove properties of all polynomial functions. (Contributed by Stefan O'Rear, 4-Oct-2014.)

**Hypotheses**
Ref	Expression
mzpindd.co	⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → 𝜒)
mzpindd.pr	⊢ ((𝜑 ∧ 𝑓 ∈ 𝑉) → 𝜃)
mzpindd.ad	⊢ ((𝜑 ∧ (𝑓:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜏) ∧ (𝑔:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜂)) → 𝜁)
mzpindd.mu	⊢ ((𝜑 ∧ (𝑓:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜏) ∧ (𝑔:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜂)) → 𝜎)
mzpindd.1	⊢ (𝑥 = ((ℤ ↑_m 𝑉) × {𝑓}) → (𝜓 ↔ 𝜒))
mzpindd.2	⊢ (𝑥 = (𝑔 ∈ (ℤ ↑_m 𝑉) ↦ (𝑔‘𝑓)) → (𝜓 ↔ 𝜃))
mzpindd.3	⊢ (𝑥 = 𝑓 → (𝜓 ↔ 𝜏))
mzpindd.4	⊢ (𝑥 = 𝑔 → (𝜓 ↔ 𝜂))
mzpindd.5	⊢ (𝑥 = (𝑓 ∘_f + 𝑔) → (𝜓 ↔ 𝜁))
mzpindd.6	⊢ (𝑥 = (𝑓 ∘_f · 𝑔) → (𝜓 ↔ 𝜎))
mzpindd.7	⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜌))

**Assertion**
Ref	Expression
mzpindd	⊢ ((𝜑 ∧ 𝐴 ∈ (mzPoly‘𝑉)) → 𝜌)

Distinct variable groups: 𝜑,𝑥,𝑓,𝑔 𝜓,𝑓,𝑔 𝜒,𝑥 𝜃,𝑥 𝜏,𝑥 𝜂,𝑥 𝜁,𝑥 𝜎,𝑥 𝜌,𝑥 𝑥,𝑉,𝑓,𝑔 𝑥,𝐴 Allowed substitution hints: 𝜓(𝑥) 𝜒(𝑓,𝑔) 𝜃(𝑓,𝑔) 𝜏(𝑓,𝑔) 𝜂(𝑓,𝑔) 𝜁(𝑓,𝑔) 𝜎(𝑓,𝑔) 𝜌(𝑓,𝑔) 𝐴(𝑓,𝑔)

Proof of Theorem mzpindd Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6928	. . . 4 ⊢ (𝐴 ∈ (mzPoly‘𝑉) → 𝑉 ∈ V)
2	1	adantl 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (mzPoly‘𝑉)) → 𝑉 ∈ V)
3		mzpval 41772	. . . . . . 7 ⊢ (𝑉 ∈ V → (mzPoly‘𝑉) = ∩ (mzPolyCld‘𝑉))
4	3	adantl 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑉 ∈ V) → (mzPoly‘𝑉) = ∩ (mzPolyCld‘𝑉))
5		ssrab2 4076	. . . . . . . . . 10 ⊢ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ⊆ (ℤ ↑_m (ℤ ↑_m 𝑉))
6	5	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑉 ∈ V) → {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ⊆ (ℤ ↑_m (ℤ ↑_m 𝑉)))
7		ovex 7444	. . . . . . . . . . . . . . 15 ⊢ (ℤ ↑_m 𝑉) ∈ V
8		zex 12571	. . . . . . . . . . . . . . 15 ⊢ ℤ ∈ V
9	7, 8	constmap 41753	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ ℤ → ((ℤ ↑_m 𝑉) × {𝑓}) ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)))
10	9	adantl 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → ((ℤ ↑_m 𝑉) × {𝑓}) ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)))
11		mzpindd.co	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → 𝜒)
12		mzpindd.1	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ((ℤ ↑_m 𝑉) × {𝑓}) → (𝜓 ↔ 𝜒))
13	12	elrab 3682	. . . . . . . . . . . . 13 ⊢ (((ℤ ↑_m 𝑉) × {𝑓}) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ↔ (((ℤ ↑_m 𝑉) × {𝑓}) ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ 𝜒))
14	10, 11, 13	sylanbrc 581	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → ((ℤ ↑_m 𝑉) × {𝑓}) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓})
15	14	ralrimiva 3144	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑓 ∈ ℤ ((ℤ ↑_m 𝑉) × {𝑓}) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓})
16	15	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑉 ∈ V) → ∀𝑓 ∈ ℤ ((ℤ ↑_m 𝑉) × {𝑓}) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓})
17	8	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑉 ∈ V) ∧ 𝑓 ∈ 𝑉) ∧ 𝑔 ∈ (ℤ ↑_m 𝑉)) → ℤ ∈ V)
18		simpllr 772	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑉 ∈ V) ∧ 𝑓 ∈ 𝑉) ∧ 𝑔 ∈ (ℤ ↑_m 𝑉)) → 𝑉 ∈ V)
19		simpr 483	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑉 ∈ V) ∧ 𝑓 ∈ 𝑉) ∧ 𝑔 ∈ (ℤ ↑_m 𝑉)) → 𝑔 ∈ (ℤ ↑_m 𝑉))
20		elmapg 8835	. . . . . . . . . . . . . . . . 17 ⊢ ((ℤ ∈ V ∧ 𝑉 ∈ V) → (𝑔 ∈ (ℤ ↑_m 𝑉) ↔ 𝑔:𝑉⟶ℤ))
21	20	biimpa 475	. . . . . . . . . . . . . . . 16 ⊢ (((ℤ ∈ V ∧ 𝑉 ∈ V) ∧ 𝑔 ∈ (ℤ ↑_m 𝑉)) → 𝑔:𝑉⟶ℤ)
22	17, 18, 19, 21	syl21anc 834	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑉 ∈ V) ∧ 𝑓 ∈ 𝑉) ∧ 𝑔 ∈ (ℤ ↑_m 𝑉)) → 𝑔:𝑉⟶ℤ)
23		simplr 765	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑉 ∈ V) ∧ 𝑓 ∈ 𝑉) ∧ 𝑔 ∈ (ℤ ↑_m 𝑉)) → 𝑓 ∈ 𝑉)
24	22, 23	ffvelcdmd 7086	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑉 ∈ V) ∧ 𝑓 ∈ 𝑉) ∧ 𝑔 ∈ (ℤ ↑_m 𝑉)) → (𝑔‘𝑓) ∈ ℤ)
25	24	fmpttd 7115	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑉 ∈ V) ∧ 𝑓 ∈ 𝑉) → (𝑔 ∈ (ℤ ↑_m 𝑉) ↦ (𝑔‘𝑓)):(ℤ ↑_m 𝑉)⟶ℤ)
26	8, 7	elmap 8867	. . . . . . . . . . . . 13 ⊢ ((𝑔 ∈ (ℤ ↑_m 𝑉) ↦ (𝑔‘𝑓)) ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ↔ (𝑔 ∈ (ℤ ↑_m 𝑉) ↦ (𝑔‘𝑓)):(ℤ ↑_m 𝑉)⟶ℤ)
27	25, 26	sylibr 233	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑉 ∈ V) ∧ 𝑓 ∈ 𝑉) → (𝑔 ∈ (ℤ ↑_m 𝑉) ↦ (𝑔‘𝑓)) ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)))
28		mzpindd.pr	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑉) → 𝜃)
29	28	adantlr 711	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑉 ∈ V) ∧ 𝑓 ∈ 𝑉) → 𝜃)
30		mzpindd.2	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑔 ∈ (ℤ ↑_m 𝑉) ↦ (𝑔‘𝑓)) → (𝜓 ↔ 𝜃))
31	30	elrab 3682	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ (ℤ ↑_m 𝑉) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ↔ ((𝑔 ∈ (ℤ ↑_m 𝑉) ↦ (𝑔‘𝑓)) ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ 𝜃))
32	27, 29, 31	sylanbrc 581	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑉 ∈ V) ∧ 𝑓 ∈ 𝑉) → (𝑔 ∈ (ℤ ↑_m 𝑉) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓})
33	32	ralrimiva 3144	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑉 ∈ V) → ∀𝑓 ∈ 𝑉 (𝑔 ∈ (ℤ ↑_m 𝑉) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓})
34	16, 33	jca 510	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑉 ∈ V) → (∀𝑓 ∈ ℤ ((ℤ ↑_m 𝑉) × {𝑓}) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ∧ ∀𝑓 ∈ 𝑉 (𝑔 ∈ (ℤ ↑_m 𝑉) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓}))
35		zaddcl 12606	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → (𝑎 + 𝑏) ∈ ℤ)
36	35	adantl 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝑔:(ℤ ↑_m 𝑉)⟶ℤ) ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → (𝑎 + 𝑏) ∈ ℤ)
37		simpl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝑔:(ℤ ↑_m 𝑉)⟶ℤ) → 𝑓:(ℤ ↑_m 𝑉)⟶ℤ)
38		simpr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝑔:(ℤ ↑_m 𝑉)⟶ℤ) → 𝑔:(ℤ ↑_m 𝑉)⟶ℤ)
39	7	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝑔:(ℤ ↑_m 𝑉)⟶ℤ) → (ℤ ↑_m 𝑉) ∈ V)
40		inidm 4217	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℤ ↑_m 𝑉) ∩ (ℤ ↑_m 𝑉)) = (ℤ ↑_m 𝑉)
41	36, 37, 38, 39, 39, 40	off 7690	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝑔:(ℤ ↑_m 𝑉)⟶ℤ) → (𝑓 ∘_f + 𝑔):(ℤ ↑_m 𝑉)⟶ℤ)
42	41	ad2ant2r 743	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜏) ∧ (𝑔:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜂)) → (𝑓 ∘_f + 𝑔):(ℤ ↑_m 𝑉)⟶ℤ)
43	42	adantl 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑓:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜏) ∧ (𝑔:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜂))) → (𝑓 ∘_f + 𝑔):(ℤ ↑_m 𝑉)⟶ℤ)
44		mzpindd.ad	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑓:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜏) ∧ (𝑔:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜂)) → 𝜁)
45	44	3expb 1118	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑓:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜏) ∧ (𝑔:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜂))) → 𝜁)
46	43, 45	jca 510	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑓:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜏) ∧ (𝑔:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜂))) → ((𝑓 ∘_f + 𝑔):(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜁))
47		zmulcl 12615	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → (𝑎 · 𝑏) ∈ ℤ)
48	47	adantl 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝑔:(ℤ ↑_m 𝑉)⟶ℤ) ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → (𝑎 · 𝑏) ∈ ℤ)
49	48, 37, 38, 39, 39, 40	off 7690	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝑔:(ℤ ↑_m 𝑉)⟶ℤ) → (𝑓 ∘_f · 𝑔):(ℤ ↑_m 𝑉)⟶ℤ)
50	49	ad2ant2r 743	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜏) ∧ (𝑔:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜂)) → (𝑓 ∘_f · 𝑔):(ℤ ↑_m 𝑉)⟶ℤ)
51	50	adantl 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑓:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜏) ∧ (𝑔:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜂))) → (𝑓 ∘_f · 𝑔):(ℤ ↑_m 𝑉)⟶ℤ)
52		mzpindd.mu	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑓:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜏) ∧ (𝑔:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜂)) → 𝜎)
53	52	3expb 1118	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑓:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜏) ∧ (𝑔:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜂))) → 𝜎)
54	46, 51, 53	jca32 514	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑓:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜏) ∧ (𝑔:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜂))) → (((𝑓 ∘_f + 𝑔):(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜁) ∧ ((𝑓 ∘_f · 𝑔):(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜎)))
55	54	ex 411	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝑓:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜏) ∧ (𝑔:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜂)) → (((𝑓 ∘_f + 𝑔):(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜁) ∧ ((𝑓 ∘_f · 𝑔):(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜎))))
56	8, 7	elmap 8867	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ↔ 𝑓:(ℤ ↑_m 𝑉)⟶ℤ)
57	56	anbi1i 622	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ 𝜏) ↔ (𝑓:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜏))
58	8, 7	elmap 8867	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ↔ 𝑔:(ℤ ↑_m 𝑉)⟶ℤ)
59	58	anbi1i 622	. . . . . . . . . . . . . 14 ⊢ ((𝑔 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ 𝜂) ↔ (𝑔:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜂))
60	57, 59	anbi12i 625	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ 𝜏) ∧ (𝑔 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ 𝜂)) ↔ ((𝑓:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜏) ∧ (𝑔:(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜂)))
61	8, 7	elmap 8867	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∘_f + 𝑔) ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ↔ (𝑓 ∘_f + 𝑔):(ℤ ↑_m 𝑉)⟶ℤ)
62	61	anbi1i 622	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∘_f + 𝑔) ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ 𝜁) ↔ ((𝑓 ∘_f + 𝑔):(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜁))
63	8, 7	elmap 8867	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∘_f · 𝑔) ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ↔ (𝑓 ∘_f · 𝑔):(ℤ ↑_m 𝑉)⟶ℤ)
64	63	anbi1i 622	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∘_f · 𝑔) ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ 𝜎) ↔ ((𝑓 ∘_f · 𝑔):(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜎))
65	62, 64	anbi12i 625	. . . . . . . . . . . . 13 ⊢ ((((𝑓 ∘_f + 𝑔) ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ 𝜁) ∧ ((𝑓 ∘_f · 𝑔) ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ 𝜎)) ↔ (((𝑓 ∘_f + 𝑔):(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜁) ∧ ((𝑓 ∘_f · 𝑔):(ℤ ↑_m 𝑉)⟶ℤ ∧ 𝜎)))
66	55, 60, 65	3imtr4g 295	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝑓 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ 𝜏) ∧ (𝑔 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ 𝜂)) → (((𝑓 ∘_f + 𝑔) ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ 𝜁) ∧ ((𝑓 ∘_f · 𝑔) ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ 𝜎))))
67		mzpindd.3	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑓 → (𝜓 ↔ 𝜏))
68	67	elrab 3682	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ↔ (𝑓 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ 𝜏))
69		mzpindd.4	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑔 → (𝜓 ↔ 𝜂))
70	69	elrab 3682	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ↔ (𝑔 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ 𝜂))
71	68, 70	anbi12i 625	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ∧ 𝑔 ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓}) ↔ ((𝑓 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ 𝜏) ∧ (𝑔 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ 𝜂)))
72		mzpindd.5	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑓 ∘_f + 𝑔) → (𝜓 ↔ 𝜁))
73	72	elrab 3682	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∘_f + 𝑔) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ↔ ((𝑓 ∘_f + 𝑔) ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ 𝜁))
74		mzpindd.6	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑓 ∘_f · 𝑔) → (𝜓 ↔ 𝜎))
75	74	elrab 3682	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∘_f · 𝑔) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ↔ ((𝑓 ∘_f · 𝑔) ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ 𝜎))
76	73, 75	anbi12i 625	. . . . . . . . . . . 12 ⊢ (((𝑓 ∘_f + 𝑔) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ∧ (𝑓 ∘_f · 𝑔) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓}) ↔ (((𝑓 ∘_f + 𝑔) ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ 𝜁) ∧ ((𝑓 ∘_f · 𝑔) ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ 𝜎)))
77	66, 71, 76	3imtr4g 295	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑓 ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ∧ 𝑔 ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓}) → ((𝑓 ∘_f + 𝑔) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ∧ (𝑓 ∘_f · 𝑔) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓})))
78	77	ralrimivv 3196	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑓 ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓}∀𝑔 ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ((𝑓 ∘_f + 𝑔) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ∧ (𝑓 ∘_f · 𝑔) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓}))
79	78	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑉 ∈ V) → ∀𝑓 ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓}∀𝑔 ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ((𝑓 ∘_f + 𝑔) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ∧ (𝑓 ∘_f · 𝑔) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓}))
80	6, 34, 79	jca32 514	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑉 ∈ V) → ({𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ⊆ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ ((∀𝑓 ∈ ℤ ((ℤ ↑_m 𝑉) × {𝑓}) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ∧ ∀𝑓 ∈ 𝑉 (𝑔 ∈ (ℤ ↑_m 𝑉) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓}) ∧ ∀𝑓 ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓}∀𝑔 ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ((𝑓 ∘_f + 𝑔) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ∧ (𝑓 ∘_f · 𝑔) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓}))))
81		elmzpcl 41766	. . . . . . . . 9 ⊢ (𝑉 ∈ V → ({𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ∈ (mzPolyCld‘𝑉) ↔ ({𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ⊆ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ ((∀𝑓 ∈ ℤ ((ℤ ↑_m 𝑉) × {𝑓}) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ∧ ∀𝑓 ∈ 𝑉 (𝑔 ∈ (ℤ ↑_m 𝑉) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓}) ∧ ∀𝑓 ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓}∀𝑔 ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ((𝑓 ∘_f + 𝑔) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ∧ (𝑓 ∘_f · 𝑔) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓})))))
82	81	adantl 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑉 ∈ V) → ({𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ∈ (mzPolyCld‘𝑉) ↔ ({𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ⊆ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ ((∀𝑓 ∈ ℤ ((ℤ ↑_m 𝑉) × {𝑓}) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ∧ ∀𝑓 ∈ 𝑉 (𝑔 ∈ (ℤ ↑_m 𝑉) ↦ (𝑔‘𝑓)) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓}) ∧ ∀𝑓 ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓}∀𝑔 ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ((𝑓 ∘_f + 𝑔) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ∧ (𝑓 ∘_f · 𝑔) ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓})))))
83	80, 82	mpbird 256	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑉 ∈ V) → {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ∈ (mzPolyCld‘𝑉))
84		intss1 4966	. . . . . . 7 ⊢ ({𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ∈ (mzPolyCld‘𝑉) → ∩ (mzPolyCld‘𝑉) ⊆ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓})
85	83, 84	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑉 ∈ V) → ∩ (mzPolyCld‘𝑉) ⊆ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓})
86	4, 85	eqsstrd 4019	. . . . 5 ⊢ ((𝜑 ∧ 𝑉 ∈ V) → (mzPoly‘𝑉) ⊆ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓})
87	86	sselda 3981	. . . 4 ⊢ (((𝜑 ∧ 𝑉 ∈ V) ∧ 𝐴 ∈ (mzPoly‘𝑉)) → 𝐴 ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓})
88	87	an32s 648	. . 3 ⊢ (((𝜑 ∧ 𝐴 ∈ (mzPoly‘𝑉)) ∧ 𝑉 ∈ V) → 𝐴 ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓})
89	2, 88	mpdan 683	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ (mzPoly‘𝑉)) → 𝐴 ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓})
90		mzpindd.7	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜌))
91	90	elrab 3682	. . 3 ⊢ (𝐴 ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} ↔ (𝐴 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∧ 𝜌))
92	91	simprbi 495	. 2 ⊢ (𝐴 ∈ {𝑥 ∈ (ℤ ↑_m (ℤ ↑_m 𝑉)) ∣ 𝜓} → 𝜌)
93	89, 92	syl 17	1 ⊢ ((𝜑 ∧ 𝐴 ∈ (mzPoly‘𝑉)) → 𝜌)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ∀wral 3059 {crab 3430 Vcvv 3472 ⊆ wss 3947 {csn 4627 ∩ cint 4949 ↦ cmpt 5230 × cxp 5673 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ∘_f cof 7670 ↑_m cmap 8822 + caddc 11115 · cmul 11117 ℤcz 12562 mzPolyCldcmzpcl 41761 mzPolycmzp 41762

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-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 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-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-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-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 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-mzpcl 41763 df-mzp 41764

This theorem is referenced by: mzpmfp 41787 mzpsubst 41788 mzpcompact2lem 41791 mzpcong 42013

W3C validator