aks6d1c5lem2 - Mathbox for metakunt

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Theorem aks6d1c5lem2 42304

Description: Lemma for Claim 5, contradiction of different evaluations that map to the same. (Contributed by metakunt, 5-May-2025.)

**Hypotheses**
Ref	Expression
aks6d1p5.1	⊢ (𝜑 → 𝐾 ∈ Field)
aks6d1p5.2	⊢ (𝜑 → 𝑃 ∈ ℙ)
aks6d1c5.3	⊢ 𝑃 = (chr‘𝐾)
aks6d1c5.4	⊢ (𝜑 → 𝐴 ∈ ℕ₀)
aks6d1c5.5	⊢ (𝜑 → 𝐴 < 𝑃)
aks6d1c5.6	⊢ 𝑋 = (var₁‘𝐾)
aks6d1c5.7	⊢ ↑ = (._g‘(mulGrp‘(Poly₁‘𝐾)))
aks6d1c5.8	⊢ 𝐺 = (𝑔 ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ ((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))
aks6d1c5p2.1	⊢ (𝜑 → 𝑌 ∈ (ℕ₀ ↑_m (0...𝐴)))
aks6d1c5p2.2	⊢ (𝜑 → 𝑍 ∈ (ℕ₀ ↑_m (0...𝐴)))
aks6d1c5p2.3	⊢ (𝜑 → (𝐺‘𝑌) = (𝐺‘𝑍))
aks6d1c5p2.4	⊢ (𝜑 → 𝑊 ∈ (0...𝐴))
aks6d1c5p2.5	⊢ (𝜑 → (𝑌‘𝑊) < (𝑍‘𝑊))

**Assertion**
Ref	Expression
aks6d1c5lem2	⊢ (𝜑 → (0_g‘𝐾) ≠ (0_g‘𝐾))

Distinct variable groups: ↑ ,𝑔,𝑖 𝐴,𝑔,𝑖 𝑔,𝐾,𝑖 𝑖,𝑊 𝑔,𝑋,𝑖 𝑔,𝑌,𝑖 𝑔,𝑍,𝑖 𝜑,𝑔,𝑖 Allowed substitution hints: 𝑃(𝑔,𝑖) 𝐺(𝑔,𝑖) 𝑊(𝑔)

**Proof of Theorem aks6d1c5lem2**
Step	Hyp	Ref	Expression
1		eqid 2733	. . . . . 6 ⊢ (eval₁‘𝐾) = (eval₁‘𝐾)
2		eqid 2733	. . . . . 6 ⊢ (Poly₁‘𝐾) = (Poly₁‘𝐾)
3		eqid 2733	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		eqid 2733	. . . . . 6 ⊢ (Base‘(Poly₁‘𝐾)) = (Base‘(Poly₁‘𝐾))
5		aks6d1p5.1	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Field)
6		isfld 20664	. . . . . . . 8 ⊢ (𝐾 ∈ Field ↔ (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing))
7	6	simprbi 496	. . . . . . 7 ⊢ (𝐾 ∈ Field → 𝐾 ∈ CRing)
8	5, 7	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ CRing)
9	8	crngringd 20172	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Ring)
10		eqid 2733	. . . . . . . . . 10 ⊢ (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
11	10	zrhrhm 21457	. . . . . . . . 9 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾))
12	9, 11	syl 17	. . . . . . . 8 ⊢ (𝜑 → (ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾))
13		zringbas 21399	. . . . . . . . 9 ⊢ ℤ = (Base‘ℤ_ring)
14	13, 3	rhmf 20411	. . . . . . . 8 ⊢ ((ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾) → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
15	12, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
16		0zd 12491	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℤ)
17		aks6d1c5p2.4	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ (0...𝐴))
18	17	elfzelzd 13432	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ ℤ)
19	16, 18	zsubcld 12592	. . . . . . 7 ⊢ (𝜑 → (0 − 𝑊) ∈ ℤ)
20	15, 19	ffvelcdmd 7027	. . . . . 6 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘(0 − 𝑊)) ∈ (Base‘𝐾))
21		eqid 2733	. . . . . . . . 9 ⊢ (mulGrp‘(Poly₁‘𝐾)) = (mulGrp‘(Poly₁‘𝐾))
22	21, 4	mgpbas 20071	. . . . . . . 8 ⊢ (Base‘(Poly₁‘𝐾)) = (Base‘(mulGrp‘(Poly₁‘𝐾)))
23		aks6d1c5.7	. . . . . . . 8 ⊢ ↑ = (._g‘(mulGrp‘(Poly₁‘𝐾)))
24	2	ply1crng 22130	. . . . . . . . . . 11 ⊢ (𝐾 ∈ CRing → (Poly₁‘𝐾) ∈ CRing)
25	8, 24	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ CRing)
26	21	crngmgp 20167	. . . . . . . . . 10 ⊢ ((Poly₁‘𝐾) ∈ CRing → (mulGrp‘(Poly₁‘𝐾)) ∈ CMnd)
27	25, 26	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (mulGrp‘(Poly₁‘𝐾)) ∈ CMnd)
28	27	cmnmndd 19724	. . . . . . . 8 ⊢ (𝜑 → (mulGrp‘(Poly₁‘𝐾)) ∈ Mnd)
29		aks6d1c5p2.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ∈ (ℕ₀ ↑_m (0...𝐴)))
30		nn0ex 12398	. . . . . . . . . . . . . . . 16 ⊢ ℕ₀ ∈ V
31	30	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ℕ₀ ∈ V)
32		ovexd 7390	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0...𝐴) ∈ V)
33		elmapg 8772	. . . . . . . . . . . . . . 15 ⊢ ((ℕ₀ ∈ V ∧ (0...𝐴) ∈ V) → (𝑌 ∈ (ℕ₀ ↑_m (0...𝐴)) ↔ 𝑌:(0...𝐴)⟶ℕ₀))
34	31, 32, 33	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑌 ∈ (ℕ₀ ↑_m (0...𝐴)) ↔ 𝑌:(0...𝐴)⟶ℕ₀))
35	29, 34	mpbid 232	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌:(0...𝐴)⟶ℕ₀)
36	35, 17	ffvelcdmd 7027	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑌‘𝑊) ∈ ℕ₀)
37	36	nn0zd 12504	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌‘𝑊) ∈ ℤ)
38	37, 37	zsubcld 12592	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑌‘𝑊) − (𝑌‘𝑊)) ∈ ℤ)
39		0red 11126	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ ℝ)
40	39	leidd 11694	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ 0)
41	36	nn0red 12454	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑌‘𝑊) ∈ ℝ)
42	41	recnd 11151	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑌‘𝑊) ∈ ℂ)
43	42	subidd 11471	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑌‘𝑊) − (𝑌‘𝑊)) = 0)
44	43	eqcomd 2739	. . . . . . . . . . 11 ⊢ (𝜑 → 0 = ((𝑌‘𝑊) − (𝑌‘𝑊)))
45	40, 44	breqtrd 5121	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ ((𝑌‘𝑊) − (𝑌‘𝑊)))
46	38, 45	jca 511	. . . . . . . . 9 ⊢ (𝜑 → (((𝑌‘𝑊) − (𝑌‘𝑊)) ∈ ℤ ∧ 0 ≤ ((𝑌‘𝑊) − (𝑌‘𝑊))))
47		elnn0z 12492	. . . . . . . . 9 ⊢ (((𝑌‘𝑊) − (𝑌‘𝑊)) ∈ ℕ₀ ↔ (((𝑌‘𝑊) − (𝑌‘𝑊)) ∈ ℤ ∧ 0 ≤ ((𝑌‘𝑊) − (𝑌‘𝑊))))
48	46, 47	sylibr 234	. . . . . . . 8 ⊢ (𝜑 → ((𝑌‘𝑊) − (𝑌‘𝑊)) ∈ ℕ₀)
49		aks6d1c5.6	. . . . . . . . . . 11 ⊢ 𝑋 = (var₁‘𝐾)
50	1, 49, 3, 2, 4, 8, 20	evl1vard 22272	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘𝑋)‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = ((ℤRHom‘𝐾)‘(0 − 𝑊))))
51		eqid 2733	. . . . . . . . . . 11 ⊢ (algSc‘(Poly₁‘𝐾)) = (algSc‘(Poly₁‘𝐾))
52	15, 18	ffvelcdmd 7027	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘𝑊) ∈ (Base‘𝐾))
53	1, 2, 3, 51, 4, 8, 52, 20	evl1scad 22270	. . . . . . . . . 10 ⊢ (𝜑 → (((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊)) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊)))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = ((ℤRHom‘𝐾)‘𝑊)))
54		eqid 2733	. . . . . . . . . 10 ⊢ (+_g‘(Poly₁‘𝐾)) = (+_g‘(Poly₁‘𝐾))
55		eqid 2733	. . . . . . . . . 10 ⊢ (+_g‘𝐾) = (+_g‘𝐾)
56	1, 2, 3, 4, 8, 20, 50, 53, 54, 55	evl1addd 22276	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘(𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = (((ℤRHom‘𝐾)‘(0 − 𝑊))(+_g‘𝐾)((ℤRHom‘𝐾)‘𝑊))))
57	56	simpld 494	. . . . . . . 8 ⊢ (𝜑 → (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))) ∈ (Base‘(Poly₁‘𝐾)))
58	22, 23, 28, 48, 57	mulgnn0cld 19016	. . . . . . 7 ⊢ (𝜑 → (((𝑌‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊)))) ∈ (Base‘(Poly₁‘𝐾)))
59	43	oveq1d 7370	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑌‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊)))) = (0 ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊)))))
60		eqid 2733	. . . . . . . . . . . . 13 ⊢ (0_g‘(mulGrp‘(Poly₁‘𝐾))) = (0_g‘(mulGrp‘(Poly₁‘𝐾)))
61	22, 60, 23	mulg0 18995	. . . . . . . . . . . 12 ⊢ ((𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))) ∈ (Base‘(Poly₁‘𝐾)) → (0 ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊)))) = (0_g‘(mulGrp‘(Poly₁‘𝐾))))
62	57, 61	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (0 ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊)))) = (0_g‘(mulGrp‘(Poly₁‘𝐾))))
63	59, 62	eqtrd 2768	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑌‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊)))) = (0_g‘(mulGrp‘(Poly₁‘𝐾))))
64	63	fveq2d 6835	. . . . . . . . 9 ⊢ (𝜑 → ((eval₁‘𝐾)‘(((𝑌‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))) = ((eval₁‘𝐾)‘(0_g‘(mulGrp‘(Poly₁‘𝐾)))))
65	64	fveq1d 6833	. . . . . . . 8 ⊢ (𝜑 → (((eval₁‘𝐾)‘(((𝑌‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊)))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = (((eval₁‘𝐾)‘(0_g‘(mulGrp‘(Poly₁‘𝐾))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))))
66		eqid 2733	. . . . . . . . . . . . . 14 ⊢ (1_r‘(Poly₁‘𝐾)) = (1_r‘(Poly₁‘𝐾))
67	21, 66	ringidval 20109	. . . . . . . . . . . . 13 ⊢ (1_r‘(Poly₁‘𝐾)) = (0_g‘(mulGrp‘(Poly₁‘𝐾)))
68	67	eqcomi 2742	. . . . . . . . . . . 12 ⊢ (0_g‘(mulGrp‘(Poly₁‘𝐾))) = (1_r‘(Poly₁‘𝐾))
69	68	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (0_g‘(mulGrp‘(Poly₁‘𝐾))) = (1_r‘(Poly₁‘𝐾)))
70	69	fveq2d 6835	. . . . . . . . . 10 ⊢ (𝜑 → ((eval₁‘𝐾)‘(0_g‘(mulGrp‘(Poly₁‘𝐾)))) = ((eval₁‘𝐾)‘(1_r‘(Poly₁‘𝐾))))
71	70	fveq1d 6833	. . . . . . . . 9 ⊢ (𝜑 → (((eval₁‘𝐾)‘(0_g‘(mulGrp‘(Poly₁‘𝐾))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = (((eval₁‘𝐾)‘(1_r‘(Poly₁‘𝐾)))‘((ℤRHom‘𝐾)‘(0 − 𝑊))))
72	2, 49, 21, 23	ply1idvr1 22229	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ Ring → (0 ↑ 𝑋) = (1_r‘(Poly₁‘𝐾)))
73	72	eqcomd 2739	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Ring → (1_r‘(Poly₁‘𝐾)) = (0 ↑ 𝑋))
74	9, 73	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (1_r‘(Poly₁‘𝐾)) = (0 ↑ 𝑋))
75	74	fveq2d 6835	. . . . . . . . . . 11 ⊢ (𝜑 → ((eval₁‘𝐾)‘(1_r‘(Poly₁‘𝐾))) = ((eval₁‘𝐾)‘(0 ↑ 𝑋)))
76	75	fveq1d 6833	. . . . . . . . . 10 ⊢ (𝜑 → (((eval₁‘𝐾)‘(1_r‘(Poly₁‘𝐾)))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = (((eval₁‘𝐾)‘(0 ↑ 𝑋))‘((ℤRHom‘𝐾)‘(0 − 𝑊))))
77		eqid 2733	. . . . . . . . . . . . 13 ⊢ (._g‘(mulGrp‘𝐾)) = (._g‘(mulGrp‘𝐾))
78	44, 48	eqeltrd 2833	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ ℕ₀)
79	1, 2, 3, 4, 8, 20, 50, 23, 77, 78	evl1expd 22280	. . . . . . . . . . . 12 ⊢ (𝜑 → ((0 ↑ 𝑋) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘(0 ↑ 𝑋))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = (0(._g‘(mulGrp‘𝐾))((ℤRHom‘𝐾)‘(0 − 𝑊)))))
80	79	simprd 495	. . . . . . . . . . 11 ⊢ (𝜑 → (((eval₁‘𝐾)‘(0 ↑ 𝑋))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = (0(._g‘(mulGrp‘𝐾))((ℤRHom‘𝐾)‘(0 − 𝑊))))
81		eqid 2733	. . . . . . . . . . . . . . . 16 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
82	81, 3	mgpbas 20071	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
83	82	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(mulGrp‘𝐾)))
84	20, 83	eleqtrd 2835	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘(0 − 𝑊)) ∈ (Base‘(mulGrp‘𝐾)))
85		eqid 2733	. . . . . . . . . . . . . 14 ⊢ (Base‘(mulGrp‘𝐾)) = (Base‘(mulGrp‘𝐾))
86		eqid 2733	. . . . . . . . . . . . . 14 ⊢ (0_g‘(mulGrp‘𝐾)) = (0_g‘(mulGrp‘𝐾))
87	85, 86, 77	mulg0 18995	. . . . . . . . . . . . 13 ⊢ (((ℤRHom‘𝐾)‘(0 − 𝑊)) ∈ (Base‘(mulGrp‘𝐾)) → (0(._g‘(mulGrp‘𝐾))((ℤRHom‘𝐾)‘(0 − 𝑊))) = (0_g‘(mulGrp‘𝐾)))
88	84, 87	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (0(._g‘(mulGrp‘𝐾))((ℤRHom‘𝐾)‘(0 − 𝑊))) = (0_g‘(mulGrp‘𝐾)))
89		eqid 2733	. . . . . . . . . . . . . . 15 ⊢ (1_r‘𝐾) = (1_r‘𝐾)
90	81, 89	ringidval 20109	. . . . . . . . . . . . . 14 ⊢ (1_r‘𝐾) = (0_g‘(mulGrp‘𝐾))
91	90	eqcomi 2742	. . . . . . . . . . . . 13 ⊢ (0_g‘(mulGrp‘𝐾)) = (1_r‘𝐾)
92	91	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (0_g‘(mulGrp‘𝐾)) = (1_r‘𝐾))
93	88, 92	eqtrd 2768	. . . . . . . . . . 11 ⊢ (𝜑 → (0(._g‘(mulGrp‘𝐾))((ℤRHom‘𝐾)‘(0 − 𝑊))) = (1_r‘𝐾))
94	80, 93	eqtrd 2768	. . . . . . . . . 10 ⊢ (𝜑 → (((eval₁‘𝐾)‘(0 ↑ 𝑋))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = (1_r‘𝐾))
95	76, 94	eqtrd 2768	. . . . . . . . 9 ⊢ (𝜑 → (((eval₁‘𝐾)‘(1_r‘(Poly₁‘𝐾)))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = (1_r‘𝐾))
96	71, 95	eqtrd 2768	. . . . . . . 8 ⊢ (𝜑 → (((eval₁‘𝐾)‘(0_g‘(mulGrp‘(Poly₁‘𝐾))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = (1_r‘𝐾))
97	65, 96	eqtrd 2768	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝐾)‘(((𝑌‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊)))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = (1_r‘𝐾))
98	58, 97	jca 511	. . . . . 6 ⊢ (𝜑 → ((((𝑌‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊)))) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘(((𝑌‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊)))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = (1_r‘𝐾)))
99		fzfid 13887	. . . . . . . . 9 ⊢ (𝜑 → (0...𝐴) ∈ Fin)
100		diffi 9095	. . . . . . . . 9 ⊢ ((0...𝐴) ∈ Fin → ((0...𝐴) ∖ {𝑊}) ∈ Fin)
101	99, 100	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((0...𝐴) ∖ {𝑊}) ∈ Fin)
102	28	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → (mulGrp‘(Poly₁‘𝐾)) ∈ Mnd)
103	35	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → 𝑌:(0...𝐴)⟶ℕ₀)
104		eldifi 4080	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) → 𝑖 ∈ (0...𝐴))
105	104	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → 𝑖 ∈ (0...𝐴))
106	103, 105	ffvelcdmd 7027	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → (𝑌‘𝑖) ∈ ℕ₀)
107	25	crngringd 20172	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ Ring)
108		ringcmn 20208	. . . . . . . . . . . . . 14 ⊢ ((Poly₁‘𝐾) ∈ Ring → (Poly₁‘𝐾) ∈ CMnd)
109	107, 108	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ CMnd)
110		cmnmnd 19717	. . . . . . . . . . . . 13 ⊢ ((Poly₁‘𝐾) ∈ CMnd → (Poly₁‘𝐾) ∈ Mnd)
111	109, 110	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ Mnd)
112	111	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → (Poly₁‘𝐾) ∈ Mnd)
113	50	simpld 494	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ (Base‘(Poly₁‘𝐾)))
114	113	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → 𝑋 ∈ (Base‘(Poly₁‘𝐾)))
115	9	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → 𝐾 ∈ Ring)
116	115, 11, 14	3syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
117	105	elfzelzd 13432	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → 𝑖 ∈ ℤ)
118	116, 117	ffvelcdmd 7027	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → ((ℤRHom‘𝐾)‘𝑖) ∈ (Base‘𝐾))
119	2, 51, 3, 4	ply1sclcl 22219	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Ring ∧ ((ℤRHom‘𝐾)‘𝑖) ∈ (Base‘𝐾)) → ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)) ∈ (Base‘(Poly₁‘𝐾)))
120	115, 118, 119	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)) ∈ (Base‘(Poly₁‘𝐾)))
121	4, 54	mndcl 18658	. . . . . . . . . . 11 ⊢ (((Poly₁‘𝐾) ∈ Mnd ∧ 𝑋 ∈ (Base‘(Poly₁‘𝐾)) ∧ ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)) ∈ (Base‘(Poly₁‘𝐾))) → (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))) ∈ (Base‘(Poly₁‘𝐾)))
122	112, 114, 120, 121	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))) ∈ (Base‘(Poly₁‘𝐾)))
123	22, 23, 102, 106, 122	mulgnn0cld 19016	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → ((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))) ∈ (Base‘(Poly₁‘𝐾)))
124	123	ralrimiva 3125	. . . . . . . 8 ⊢ (𝜑 → ∀𝑖 ∈ ((0...𝐴) ∖ {𝑊})((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))) ∈ (Base‘(Poly₁‘𝐾)))
125	22, 27, 101, 124	gsummptcl 19887	. . . . . . 7 ⊢ (𝜑 → ((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))) ∈ (Base‘(Poly₁‘𝐾)))
126	124	r19.21bi 3225	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → ((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))) ∈ (Base‘(Poly₁‘𝐾)))
127	126	ralrimiva 3125	. . . . . . . 8 ⊢ (𝜑 → ∀𝑖 ∈ ((0...𝐴) ∖ {𝑊})((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))) ∈ (Base‘(Poly₁‘𝐾)))
128	1, 2, 21, 3, 4, 81, 8, 20, 127, 101	evl1gprodd 42283	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝐾)‘((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = ((mulGrp‘𝐾) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ (((eval₁‘𝐾)‘((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))))))
129	125, 128	jca 511	. . . . . 6 ⊢ (𝜑 → (((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = ((mulGrp‘𝐾) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ (((eval₁‘𝐾)‘((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))‘((ℤRHom‘𝐾)‘(0 − 𝑊)))))))
130		eqid 2733	. . . . . . . 8 ⊢ (._r‘(Poly₁‘𝐾)) = (._r‘(Poly₁‘𝐾))
131	21, 130	mgpplusg 20070	. . . . . . 7 ⊢ (._r‘(Poly₁‘𝐾)) = (+_g‘(mulGrp‘(Poly₁‘𝐾)))
132	131	eqcomi 2742	. . . . . 6 ⊢ (+_g‘(mulGrp‘(Poly₁‘𝐾))) = (._r‘(Poly₁‘𝐾))
133		eqid 2733	. . . . . 6 ⊢ (._r‘𝐾) = (._r‘𝐾)
134	1, 2, 3, 4, 8, 20, 98, 129, 132, 133	evl1muld 22278	. . . . 5 ⊢ (𝜑 → (((((𝑌‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))(+_g‘(mulGrp‘(Poly₁‘𝐾)))((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘((((𝑌‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))(+_g‘(mulGrp‘(Poly₁‘𝐾)))((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = ((1_r‘𝐾)(._r‘𝐾)((mulGrp‘𝐾) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ (((eval₁‘𝐾)‘((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))))))))
135	134	simprd 495	. . . 4 ⊢ (𝜑 → (((eval₁‘𝐾)‘((((𝑌‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))(+_g‘(mulGrp‘(Poly₁‘𝐾)))((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = ((1_r‘𝐾)(._r‘𝐾)((mulGrp‘𝐾) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ (((eval₁‘𝐾)‘((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))‘((ℤRHom‘𝐾)‘(0 − 𝑊)))))))
136		fldidom 20695	. . . . . . . 8 ⊢ (𝐾 ∈ Field → 𝐾 ∈ IDomn)
137	5, 136	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ IDomn)
138		isidom 20649	. . . . . . 7 ⊢ (𝐾 ∈ IDomn ↔ (𝐾 ∈ CRing ∧ 𝐾 ∈ Domn))
139	137, 138	sylib 218	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ CRing ∧ 𝐾 ∈ Domn))
140	139	simprd 495	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Domn)
141	90	a1i 11	. . . . . . 7 ⊢ (𝜑 → (1_r‘𝐾) = (0_g‘(mulGrp‘𝐾)))
142	81	ringmgp 20165	. . . . . . . . 9 ⊢ (𝐾 ∈ Ring → (mulGrp‘𝐾) ∈ Mnd)
143	9, 142	syl 17	. . . . . . . 8 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ Mnd)
144	82, 86	mndidcl 18665	. . . . . . . 8 ⊢ ((mulGrp‘𝐾) ∈ Mnd → (0_g‘(mulGrp‘𝐾)) ∈ (Base‘𝐾))
145	143, 144	syl 17	. . . . . . 7 ⊢ (𝜑 → (0_g‘(mulGrp‘𝐾)) ∈ (Base‘𝐾))
146	141, 145	eqeltrd 2833	. . . . . 6 ⊢ (𝜑 → (1_r‘𝐾) ∈ (Base‘𝐾))
147	5	flddrngd 20665	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ DivRing)
148		eqid 2733	. . . . . . . 8 ⊢ (0_g‘𝐾) = (0_g‘𝐾)
149	148, 89	drngunz 20671	. . . . . . 7 ⊢ (𝐾 ∈ DivRing → (1_r‘𝐾) ≠ (0_g‘𝐾))
150	147, 149	syl 17	. . . . . 6 ⊢ (𝜑 → (1_r‘𝐾) ≠ (0_g‘𝐾))
151	146, 150	jca 511	. . . . 5 ⊢ (𝜑 → ((1_r‘𝐾) ∈ (Base‘𝐾) ∧ (1_r‘𝐾) ≠ (0_g‘𝐾)))
152	81	crngmgp 20167	. . . . . . . 8 ⊢ (𝐾 ∈ CRing → (mulGrp‘𝐾) ∈ CMnd)
153	8, 152	syl 17	. . . . . . 7 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ CMnd)
154	8	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → 𝐾 ∈ CRing)
155	20	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → ((ℤRHom‘𝐾)‘(0 − 𝑊)) ∈ (Base‘𝐾))
156	1, 2, 3, 4, 154, 155, 123	fveval1fvcl 22268	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → (((eval₁‘𝐾)‘((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) ∈ (Base‘𝐾))
157	156	ralrimiva 3125	. . . . . . 7 ⊢ (𝜑 → ∀𝑖 ∈ ((0...𝐴) ∖ {𝑊})(((eval₁‘𝐾)‘((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) ∈ (Base‘𝐾))
158	82, 153, 101, 157	gsummptcl 19887	. . . . . 6 ⊢ (𝜑 → ((mulGrp‘𝐾) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ (((eval₁‘𝐾)‘((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))))) ∈ (Base‘𝐾))
159	22	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → (Base‘(Poly₁‘𝐾)) = (Base‘(mulGrp‘(Poly₁‘𝐾))))
160	122, 159	eleqtrd 2835	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))) ∈ (Base‘(mulGrp‘(Poly₁‘𝐾))))
161	22	eqcomi 2742	. . . . . . . . . . . . 13 ⊢ (Base‘(mulGrp‘(Poly₁‘𝐾))) = (Base‘(Poly₁‘𝐾))
162	161	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → (Base‘(mulGrp‘(Poly₁‘𝐾))) = (Base‘(Poly₁‘𝐾)))
163	160, 162	eleqtrd 2835	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))) ∈ (Base‘(Poly₁‘𝐾)))
164		eqidd 2734	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → (((eval₁‘𝐾)‘(𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = (((eval₁‘𝐾)‘(𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))))
165	163, 164	jca 511	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → ((𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘(𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = (((eval₁‘𝐾)‘(𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))‘((ℤRHom‘𝐾)‘(0 − 𝑊)))))
166	1, 2, 3, 4, 154, 155, 165, 23, 77, 106	evl1expd 22280	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → (((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = ((𝑌‘𝑖)(._g‘(mulGrp‘𝐾))(((eval₁‘𝐾)‘(𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))))))
167	166	simprd 495	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → (((eval₁‘𝐾)‘((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = ((𝑌‘𝑖)(._g‘(mulGrp‘𝐾))(((eval₁‘𝐾)‘(𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))‘((ℤRHom‘𝐾)‘(0 − 𝑊)))))
168	137	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → 𝐾 ∈ IDomn)
169	1, 2, 3, 4, 154, 155, 163	fveval1fvcl 22268	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → (((eval₁‘𝐾)‘(𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) ∈ (Base‘𝐾))
170		eldifsni 4743	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) → 𝑖 ≠ 𝑊)
171	170	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → 𝑖 ≠ 𝑊)
172	5	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → 𝐾 ∈ Field)
173		aks6d1p5.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ ℙ)
174	173	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → 𝑃 ∈ ℙ)
175		aks6d1c5.3	. . . . . . . . . . . 12 ⊢ 𝑃 = (chr‘𝐾)
176		aks6d1c5.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℕ₀)
177	176	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → 𝐴 ∈ ℕ₀)
178		aks6d1c5.5	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 < 𝑃)
179	178	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → 𝐴 < 𝑃)
180		aks6d1c5.8	. . . . . . . . . . . 12 ⊢ 𝐺 = (𝑔 ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ ((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))
181	17	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → 𝑊 ∈ (0...𝐴))
182	172, 174, 175, 177, 179, 49, 23, 180, 105, 181	aks6d1c5lem1 42302	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → (𝑖 = 𝑊 ↔ (((eval₁‘𝐾)‘(𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = (0_g‘𝐾)))
183	182	necon3bid 2973	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → (𝑖 ≠ 𝑊 ↔ (((eval₁‘𝐾)‘(𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) ≠ (0_g‘𝐾)))
184	171, 183	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → (((eval₁‘𝐾)‘(𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) ≠ (0_g‘𝐾))
185	168, 169, 184, 106, 77	idomnnzpownz 42298	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → ((𝑌‘𝑖)(._g‘(mulGrp‘𝐾))(((eval₁‘𝐾)‘(𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))‘((ℤRHom‘𝐾)‘(0 − 𝑊)))) ≠ (0_g‘𝐾))
186	167, 185	eqnetrd 2996	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → (((eval₁‘𝐾)‘((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) ≠ (0_g‘𝐾))
187	81, 137, 101, 156, 186	idomnnzgmulnz 42299	. . . . . 6 ⊢ (𝜑 → ((mulGrp‘𝐾) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ (((eval₁‘𝐾)‘((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))))) ≠ (0_g‘𝐾))
188	158, 187	jca 511	. . . . 5 ⊢ (𝜑 → (((mulGrp‘𝐾) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ (((eval₁‘𝐾)‘((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))))) ∈ (Base‘𝐾) ∧ ((mulGrp‘𝐾) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ (((eval₁‘𝐾)‘((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))))) ≠ (0_g‘𝐾)))
189	3, 133, 148	domnmuln0 20633	. . . . 5 ⊢ ((𝐾 ∈ Domn ∧ ((1_r‘𝐾) ∈ (Base‘𝐾) ∧ (1_r‘𝐾) ≠ (0_g‘𝐾)) ∧ (((mulGrp‘𝐾) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ (((eval₁‘𝐾)‘((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))))) ∈ (Base‘𝐾) ∧ ((mulGrp‘𝐾) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ (((eval₁‘𝐾)‘((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))))) ≠ (0_g‘𝐾))) → ((1_r‘𝐾)(._r‘𝐾)((mulGrp‘𝐾) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ (((eval₁‘𝐾)‘((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))‘((ℤRHom‘𝐾)‘(0 − 𝑊)))))) ≠ (0_g‘𝐾))
190	140, 151, 188, 189	syl3anc 1373	. . . 4 ⊢ (𝜑 → ((1_r‘𝐾)(._r‘𝐾)((mulGrp‘𝐾) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ (((eval₁‘𝐾)‘((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))‘((ℤRHom‘𝐾)‘(0 − 𝑊)))))) ≠ (0_g‘𝐾))
191	135, 190	eqnetrd 2996	. . 3 ⊢ (𝜑 → (((eval₁‘𝐾)‘((((𝑌‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))(+_g‘(mulGrp‘(Poly₁‘𝐾)))((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) ≠ (0_g‘𝐾))
192	191	necomd 2984	. 2 ⊢ (𝜑 → (0_g‘𝐾) ≠ (((eval₁‘𝐾)‘((((𝑌‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))(+_g‘(mulGrp‘(Poly₁‘𝐾)))((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))))
193	41	leidd 11694	. . . . . . . 8 ⊢ (𝜑 → (𝑌‘𝑊) ≤ (𝑌‘𝑊))
194		eqid 2733	. . . . . . . 8 ⊢ (quot_1p‘𝐾) = (quot_1p‘𝐾)
195	5, 173, 175, 176, 178, 49, 23, 180, 29, 17, 36, 193, 194, 51, 21	aks6d1c5lem3 42303	. . . . . . 7 ⊢ (𝜑 → ((𝐺‘𝑌)(quot_1p‘𝐾)((𝑌‘𝑊) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))) = ((((𝑌‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))(+_g‘(mulGrp‘(Poly₁‘𝐾)))((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))))
196	195	eqcomd 2739	. . . . . 6 ⊢ (𝜑 → ((((𝑌‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))(+_g‘(mulGrp‘(Poly₁‘𝐾)))((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))) = ((𝐺‘𝑌)(quot_1p‘𝐾)((𝑌‘𝑊) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))))
197		aks6d1c5p2.3	. . . . . . 7 ⊢ (𝜑 → (𝐺‘𝑌) = (𝐺‘𝑍))
198	197	oveq1d 7370	. . . . . 6 ⊢ (𝜑 → ((𝐺‘𝑌)(quot_1p‘𝐾)((𝑌‘𝑊) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))) = ((𝐺‘𝑍)(quot_1p‘𝐾)((𝑌‘𝑊) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))))
199		aks6d1c5p2.2	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (ℕ₀ ↑_m (0...𝐴)))
200		elmapg 8772	. . . . . . . . . . . 12 ⊢ ((ℕ₀ ∈ V ∧ (0...𝐴) ∈ V) → (𝑍 ∈ (ℕ₀ ↑_m (0...𝐴)) ↔ 𝑍:(0...𝐴)⟶ℕ₀))
201	31, 32, 200	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑍 ∈ (ℕ₀ ↑_m (0...𝐴)) ↔ 𝑍:(0...𝐴)⟶ℕ₀))
202	199, 201	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍:(0...𝐴)⟶ℕ₀)
203	202, 17	ffvelcdmd 7027	. . . . . . . . 9 ⊢ (𝜑 → (𝑍‘𝑊) ∈ ℕ₀)
204	203	nn0red 12454	. . . . . . . 8 ⊢ (𝜑 → (𝑍‘𝑊) ∈ ℝ)
205		aks6d1c5p2.5	. . . . . . . 8 ⊢ (𝜑 → (𝑌‘𝑊) < (𝑍‘𝑊))
206	41, 204, 205	ltled 11272	. . . . . . 7 ⊢ (𝜑 → (𝑌‘𝑊) ≤ (𝑍‘𝑊))
207	5, 173, 175, 176, 178, 49, 23, 180, 199, 17, 36, 206, 194, 51, 21	aks6d1c5lem3 42303	. . . . . 6 ⊢ (𝜑 → ((𝐺‘𝑍)(quot_1p‘𝐾)((𝑌‘𝑊) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))) = ((((𝑍‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))(+_g‘(mulGrp‘(Poly₁‘𝐾)))((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑍‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))))
208	196, 198, 207	3eqtrd 2772	. . . . 5 ⊢ (𝜑 → ((((𝑌‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))(+_g‘(mulGrp‘(Poly₁‘𝐾)))((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))) = ((((𝑍‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))(+_g‘(mulGrp‘(Poly₁‘𝐾)))((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑍‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))))
209	208	fveq2d 6835	. . . 4 ⊢ (𝜑 → ((eval₁‘𝐾)‘((((𝑌‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))(+_g‘(mulGrp‘(Poly₁‘𝐾)))((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))) = ((eval₁‘𝐾)‘((((𝑍‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))(+_g‘(mulGrp‘(Poly₁‘𝐾)))((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑍‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))))
210	209	fveq1d 6833	. . 3 ⊢ (𝜑 → (((eval₁‘𝐾)‘((((𝑌‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))(+_g‘(mulGrp‘(Poly₁‘𝐾)))((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = (((eval₁‘𝐾)‘((((𝑍‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))(+_g‘(mulGrp‘(Poly₁‘𝐾)))((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑍‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))))
211	203	nn0zd 12504	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑍‘𝑊) ∈ ℤ)
212	211, 37	zsubcld 12592	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑍‘𝑊) − (𝑌‘𝑊)) ∈ ℤ)
213	204, 41	resubcld 11556	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑍‘𝑊) − (𝑌‘𝑊)) ∈ ℝ)
214	41, 204	posdifd 11715	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑌‘𝑊) < (𝑍‘𝑊) ↔ 0 < ((𝑍‘𝑊) − (𝑌‘𝑊))))
215	205, 214	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < ((𝑍‘𝑊) − (𝑌‘𝑊)))
216	39, 213, 215	ltled 11272	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ ((𝑍‘𝑊) − (𝑌‘𝑊)))
217	212, 216	jca 511	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑍‘𝑊) − (𝑌‘𝑊)) ∈ ℤ ∧ 0 ≤ ((𝑍‘𝑊) − (𝑌‘𝑊))))
218		elnn0z 12492	. . . . . . . . . 10 ⊢ (((𝑍‘𝑊) − (𝑌‘𝑊)) ∈ ℕ₀ ↔ (((𝑍‘𝑊) − (𝑌‘𝑊)) ∈ ℤ ∧ 0 ≤ ((𝑍‘𝑊) − (𝑌‘𝑊))))
219	217, 218	sylibr 234	. . . . . . . . 9 ⊢ (𝜑 → ((𝑍‘𝑊) − (𝑌‘𝑊)) ∈ ℕ₀)
220	1, 2, 3, 4, 8, 20, 56, 23, 77, 219	evl1expd 22280	. . . . . . . 8 ⊢ (𝜑 → ((((𝑍‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊)))) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘(((𝑍‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊)))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = (((𝑍‘𝑊) − (𝑌‘𝑊))(._g‘(mulGrp‘𝐾))(((ℤRHom‘𝐾)‘(0 − 𝑊))(+_g‘𝐾)((ℤRHom‘𝐾)‘𝑊)))))
221	220	simpld 494	. . . . . . 7 ⊢ (𝜑 → (((𝑍‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊)))) ∈ (Base‘(Poly₁‘𝐾)))
222	220	simprd 495	. . . . . . . 8 ⊢ (𝜑 → (((eval₁‘𝐾)‘(((𝑍‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊)))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = (((𝑍‘𝑊) − (𝑌‘𝑊))(._g‘(mulGrp‘𝐾))(((ℤRHom‘𝐾)‘(0 − 𝑊))(+_g‘𝐾)((ℤRHom‘𝐾)‘𝑊))))
223		rhmghm 20410	. . . . . . . . . . . . 13 ⊢ ((ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾) → (ℤRHom‘𝐾) ∈ (ℤ_ring GrpHom 𝐾))
224	12, 223	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℤRHom‘𝐾) ∈ (ℤ_ring GrpHom 𝐾))
225	19, 13	eleqtrdi 2843	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 − 𝑊) ∈ (Base‘ℤ_ring))
226	18, 13	eleqtrdi 2843	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ∈ (Base‘ℤ_ring))
227		eqid 2733	. . . . . . . . . . . . 13 ⊢ (Base‘ℤ_ring) = (Base‘ℤ_ring)
228		eqid 2733	. . . . . . . . . . . . 13 ⊢ (+_g‘ℤ_ring) = (+_g‘ℤ_ring)
229	227, 228, 55	ghmlin 19141	. . . . . . . . . . . 12 ⊢ (((ℤRHom‘𝐾) ∈ (ℤ_ring GrpHom 𝐾) ∧ (0 − 𝑊) ∈ (Base‘ℤ_ring) ∧ 𝑊 ∈ (Base‘ℤ_ring)) → ((ℤRHom‘𝐾)‘((0 − 𝑊)(+_g‘ℤ_ring)𝑊)) = (((ℤRHom‘𝐾)‘(0 − 𝑊))(+_g‘𝐾)((ℤRHom‘𝐾)‘𝑊)))
230	224, 225, 226, 229	syl3anc 1373	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘((0 − 𝑊)(+_g‘ℤ_ring)𝑊)) = (((ℤRHom‘𝐾)‘(0 − 𝑊))(+_g‘𝐾)((ℤRHom‘𝐾)‘𝑊)))
231		zringplusg 21400	. . . . . . . . . . . . . . . . 17 ⊢ + = (+_g‘ℤ_ring)
232	231	eqcomi 2742	. . . . . . . . . . . . . . . 16 ⊢ (+_g‘ℤ_ring) = +
233	232	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (+_g‘ℤ_ring) = + )
234	233	oveqd 7372	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((0 − 𝑊)(+_g‘ℤ_ring)𝑊) = ((0 − 𝑊) + 𝑊))
235	234	fveq2d 6835	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘((0 − 𝑊)(+_g‘ℤ_ring)𝑊)) = ((ℤRHom‘𝐾)‘((0 − 𝑊) + 𝑊)))
236		0cnd 11116	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℂ)
237	18	zcnd 12588	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑊 ∈ ℂ)
238	236, 237	npcand 11487	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((0 − 𝑊) + 𝑊) = 0)
239	238	fveq2d 6835	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘((0 − 𝑊) + 𝑊)) = ((ℤRHom‘𝐾)‘0))
240	235, 239	eqtrd 2768	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘((0 − 𝑊)(+_g‘ℤ_ring)𝑊)) = ((ℤRHom‘𝐾)‘0))
241	10, 148	zrh0 21459	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Ring → ((ℤRHom‘𝐾)‘0) = (0_g‘𝐾))
242	9, 241	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘0) = (0_g‘𝐾))
243	240, 242	eqtrd 2768	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℤRHom‘𝐾)‘((0 − 𝑊)(+_g‘ℤ_ring)𝑊)) = (0_g‘𝐾))
244	230, 243	eqtr3d 2770	. . . . . . . . . 10 ⊢ (𝜑 → (((ℤRHom‘𝐾)‘(0 − 𝑊))(+_g‘𝐾)((ℤRHom‘𝐾)‘𝑊)) = (0_g‘𝐾))
245	244	oveq2d 7371	. . . . . . . . 9 ⊢ (𝜑 → (((𝑍‘𝑊) − (𝑌‘𝑊))(._g‘(mulGrp‘𝐾))(((ℤRHom‘𝐾)‘(0 − 𝑊))(+_g‘𝐾)((ℤRHom‘𝐾)‘𝑊))) = (((𝑍‘𝑊) − (𝑌‘𝑊))(._g‘(mulGrp‘𝐾))(0_g‘𝐾)))
246	219	nn0zd 12504	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑍‘𝑊) − (𝑌‘𝑊)) ∈ ℤ)
247	246, 215	jca 511	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑍‘𝑊) − (𝑌‘𝑊)) ∈ ℤ ∧ 0 < ((𝑍‘𝑊) − (𝑌‘𝑊))))
248		elnnz 12489	. . . . . . . . . . 11 ⊢ (((𝑍‘𝑊) − (𝑌‘𝑊)) ∈ ℕ ↔ (((𝑍‘𝑊) − (𝑌‘𝑊)) ∈ ℤ ∧ 0 < ((𝑍‘𝑊) − (𝑌‘𝑊))))
249	247, 248	sylibr 234	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑍‘𝑊) − (𝑌‘𝑊)) ∈ ℕ)
250	9, 249, 77	ringexp0nn 42300	. . . . . . . . 9 ⊢ (𝜑 → (((𝑍‘𝑊) − (𝑌‘𝑊))(._g‘(mulGrp‘𝐾))(0_g‘𝐾)) = (0_g‘𝐾))
251	245, 250	eqtrd 2768	. . . . . . . 8 ⊢ (𝜑 → (((𝑍‘𝑊) − (𝑌‘𝑊))(._g‘(mulGrp‘𝐾))(((ℤRHom‘𝐾)‘(0 − 𝑊))(+_g‘𝐾)((ℤRHom‘𝐾)‘𝑊))) = (0_g‘𝐾))
252	222, 251	eqtrd 2768	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝐾)‘(((𝑍‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊)))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = (0_g‘𝐾))
253	221, 252	jca 511	. . . . . 6 ⊢ (𝜑 → ((((𝑍‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊)))) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘(((𝑍‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊)))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = (0_g‘𝐾)))
254		eqid 2733	. . . . . . . . . . 11 ⊢ (Base‘(mulGrp‘(Poly₁‘𝐾))) = (Base‘(mulGrp‘(Poly₁‘𝐾)))
255	202	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → 𝑍:(0...𝐴)⟶ℕ₀)
256	255, 105	ffvelcdmd 7027	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → (𝑍‘𝑖) ∈ ℕ₀)
257	254, 23, 102, 256, 160	mulgnn0cld 19016	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → ((𝑍‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))) ∈ (Base‘(mulGrp‘(Poly₁‘𝐾))))
258	257, 162	eleqtrd 2835	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ((0...𝐴) ∖ {𝑊})) → ((𝑍‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))) ∈ (Base‘(Poly₁‘𝐾)))
259	258	ralrimiva 3125	. . . . . . . 8 ⊢ (𝜑 → ∀𝑖 ∈ ((0...𝐴) ∖ {𝑊})((𝑍‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))) ∈ (Base‘(Poly₁‘𝐾)))
260	22, 27, 101, 259	gsummptcl 19887	. . . . . . 7 ⊢ (𝜑 → ((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑍‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))) ∈ (Base‘(Poly₁‘𝐾)))
261		eqidd 2734	. . . . . . 7 ⊢ (𝜑 → (((eval₁‘𝐾)‘((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑍‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = (((eval₁‘𝐾)‘((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑍‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))))
262	260, 261	jca 511	. . . . . 6 ⊢ (𝜑 → (((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑍‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑍‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = (((eval₁‘𝐾)‘((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑍‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))‘((ℤRHom‘𝐾)‘(0 − 𝑊)))))
263	1, 2, 3, 4, 8, 20, 253, 262, 132, 133	evl1muld 22278	. . . . 5 ⊢ (𝜑 → (((((𝑍‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))(+_g‘(mulGrp‘(Poly₁‘𝐾)))((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑍‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))) ∈ (Base‘(Poly₁‘𝐾)) ∧ (((eval₁‘𝐾)‘((((𝑍‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))(+_g‘(mulGrp‘(Poly₁‘𝐾)))((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑍‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = ((0_g‘𝐾)(._r‘𝐾)(((eval₁‘𝐾)‘((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑍‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))))))
264	263	simprd 495	. . . 4 ⊢ (𝜑 → (((eval₁‘𝐾)‘((((𝑍‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))(+_g‘(mulGrp‘(Poly₁‘𝐾)))((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑍‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = ((0_g‘𝐾)(._r‘𝐾)(((eval₁‘𝐾)‘((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑍‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))‘((ℤRHom‘𝐾)‘(0 − 𝑊)))))
265	1, 2, 3, 4, 8, 20, 260	fveval1fvcl 22268	. . . . 5 ⊢ (𝜑 → (((eval₁‘𝐾)‘((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑍‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) ∈ (Base‘𝐾))
266	3, 133, 148, 9, 265	ringlzd 20221	. . . 4 ⊢ (𝜑 → ((0_g‘𝐾)(._r‘𝐾)(((eval₁‘𝐾)‘((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑍‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))‘((ℤRHom‘𝐾)‘(0 − 𝑊)))) = (0_g‘𝐾))
267	264, 266	eqtrd 2768	. . 3 ⊢ (𝜑 → (((eval₁‘𝐾)‘((((𝑍‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))(+_g‘(mulGrp‘(Poly₁‘𝐾)))((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑍‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = (0_g‘𝐾))
268	210, 267	eqtrd 2768	. 2 ⊢ (𝜑 → (((eval₁‘𝐾)‘((((𝑌‘𝑊) − (𝑌‘𝑊)) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑊))))(+_g‘(mulGrp‘(Poly₁‘𝐾)))((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))))‘((ℤRHom‘𝐾)‘(0 − 𝑊))) = (0_g‘𝐾))
269	192, 268	neeqtrd 2998	1 ⊢ (𝜑 → (0_g‘𝐾) ≠ (0_g‘𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2929 Vcvv 3437 ∖ cdif 3895 {csn 4577 class class class wbr 5095 ↦ cmpt 5176 ⟶wf 6485 ‘cfv 6489 (class class class)co 7355 ↑_m cmap 8759 Fincfn 8879 0cc0 11017 + caddc 11020 < clt 11157 ≤ cle 11158 − cmin 11355 ℕcn 12136 ℕ₀cn0 12392 ℤcz 12479 ...cfz 13414 ℙcprime 16589 Basecbs 17127 +_gcplusg 17168 ._rcmulr 17169 0_gc0g 17350 Σ_g cgsu 17351 Mndcmnd 18650 ._gcmg 18988 GrpHom cghm 19132 CMndccmn 19700 mulGrpcmgp 20066 1_rcur 20107 Ringcrg 20159 CRingccrg 20160 RingHom crh 20396 Domncdomn 20616 IDomncidom 20617 DivRingcdr 20653 Fieldcfield 20654 ℤ_ringczring 21392 ℤRHomczrh 21445 chrcchr 21447 algSccascl 21798 var₁cv1 22107 Poly₁cpl1 22108 eval₁ce1 22249 quot_1pcq1p 26080

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-pre-sup 11095 ax-addf 11096 ax-mulf 11097

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-of 7619 df-ofr 7620 df-om 7806 df-1st 7930 df-2nd 7931 df-supp 8100 df-tpos 8165 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-er 8631 df-map 8761 df-pm 8762 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9257 df-sup 9337 df-inf 9338 df-oi 9407 df-card 9843 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-div 11786 df-nn 12137 df-2 12199 df-3 12200 df-4 12201 df-5 12202 df-6 12203 df-7 12204 df-8 12205 df-9 12206 df-n0 12393 df-z 12480 df-dec 12599 df-uz 12743 df-rp 12897 df-fz 13415 df-fzo 13562 df-fl 13703 df-mod 13781 df-seq 13916 df-exp 13976 df-hash 14245 df-cj 15013 df-re 15014 df-im 15015 df-sqrt 15149 df-abs 15150 df-dvds 16171 df-prm 16590 df-struct 17065 df-sets 17082 df-slot 17100 df-ndx 17112 df-base 17128 df-ress 17149 df-plusg 17181 df-mulr 17182 df-starv 17183 df-sca 17184 df-vsca 17185 df-ip 17186 df-tset 17187 df-ple 17188 df-ds 17190 df-unif 17191 df-hom 17192 df-cco 17193 df-0g 17352 df-gsum 17353 df-prds 17358 df-pws 17360 df-mre 17496 df-mrc 17497 df-acs 17499 df-mgm 18556 df-sgrp 18635 df-mnd 18651 df-mhm 18699 df-submnd 18700 df-grp 18857 df-minusg 18858 df-sbg 18859 df-mulg 18989 df-subg 19044 df-ghm 19133 df-cntz 19237 df-od 19448 df-cmn 19702 df-abl 19703 df-mgp 20067 df-rng 20079 df-ur 20108 df-srg 20113 df-ring 20161 df-cring 20162 df-oppr 20264 df-dvdsr 20284 df-unit 20285 df-invr 20315 df-rhm 20399 df-nzr 20437 df-subrng 20470 df-subrg 20494 df-rlreg 20618 df-domn 20619 df-idom 20620 df-drng 20655 df-field 20656 df-lmod 20804 df-lss 20874 df-lsp 20914 df-cnfld 21301 df-zring 21393 df-zrh 21449 df-chr 21451 df-assa 21799 df-asp 21800 df-ascl 21801 df-psr 21856 df-mvr 21857 df-mpl 21858 df-opsr 21860 df-evls 22020 df-evl 22021 df-psr1 22111 df-vr1 22112 df-ply1 22113 df-coe1 22114 df-evl1 22251 df-mdeg 26007 df-deg1 26008 df-uc1p 26084 df-q1p 26085

This theorem is referenced by: aks6d1c5 42305

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