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Theorem fta1blem 25678

Description: Lemma for fta1b 25679. (Contributed by Mario Carneiro, 14-Jun-2015.)

**Hypotheses**
Ref	Expression
fta1b.p	⊢ 𝑃 = (Poly₁‘𝑅)
fta1b.b	⊢ 𝐵 = (Base‘𝑃)
fta1b.d	⊢ 𝐷 = ( deg₁ ‘𝑅)
fta1b.o	⊢ 𝑂 = (eval₁‘𝑅)
fta1b.w	⊢ 𝑊 = (0_g‘𝑅)
fta1b.z	⊢ 0 = (0_g‘𝑃)
fta1blem.k	⊢ 𝐾 = (Base‘𝑅)
fta1blem.t	⊢ × = (._r‘𝑅)
fta1blem.x	⊢ 𝑋 = (var₁‘𝑅)
fta1blem.s	⊢ · = ( ·_𝑠 ‘𝑃)
fta1blem.1	⊢ (𝜑 → 𝑅 ∈ CRing)
fta1blem.2	⊢ (𝜑 → 𝑀 ∈ 𝐾)
fta1blem.3	⊢ (𝜑 → 𝑁 ∈ 𝐾)
fta1blem.4	⊢ (𝜑 → (𝑀 × 𝑁) = 𝑊)
fta1blem.5	⊢ (𝜑 → 𝑀 ≠ 𝑊)
fta1blem.6	⊢ (𝜑 → ((𝑀 · 𝑋) ∈ (𝐵 ∖ { 0 }) → (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ≤ (𝐷‘(𝑀 · 𝑋))))

**Assertion**
Ref	Expression
fta1blem	⊢ (𝜑 → 𝑁 = 𝑊)

**Proof of Theorem fta1blem**
Step	Hyp	Ref	Expression
1		fta1blem.3	. . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝐾)
2		fta1b.o	. . . . . . 7 ⊢ 𝑂 = (eval₁‘𝑅)
3		fta1b.p	. . . . . . 7 ⊢ 𝑃 = (Poly₁‘𝑅)
4		fta1blem.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝑅)
5		fta1b.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑃)
6		fta1blem.1	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ CRing)
7		fta1blem.x	. . . . . . . 8 ⊢ 𝑋 = (var₁‘𝑅)
8	2, 7, 4, 3, 5, 6, 1	evl1vard 21848	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑋)‘𝑁) = 𝑁))
9		fta1blem.2	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ 𝐾)
10		fta1blem.s	. . . . . . 7 ⊢ · = ( ·_𝑠 ‘𝑃)
11		fta1blem.t	. . . . . . 7 ⊢ × = (._r‘𝑅)
12	2, 3, 4, 5, 6, 1, 8, 9, 10, 11	evl1vsd 21855	. . . . . 6 ⊢ (𝜑 → ((𝑀 · 𝑋) ∈ 𝐵 ∧ ((𝑂‘(𝑀 · 𝑋))‘𝑁) = (𝑀 × 𝑁)))
13	12	simprd 497	. . . . 5 ⊢ (𝜑 → ((𝑂‘(𝑀 · 𝑋))‘𝑁) = (𝑀 × 𝑁))
14		fta1blem.4	. . . . 5 ⊢ (𝜑 → (𝑀 × 𝑁) = 𝑊)
15	13, 14	eqtrd 2773	. . . 4 ⊢ (𝜑 → ((𝑂‘(𝑀 · 𝑋))‘𝑁) = 𝑊)
16		eqid 2733	. . . . . . 7 ⊢ (𝑅 ↑_s 𝐾) = (𝑅 ↑_s 𝐾)
17		eqid 2733	. . . . . . 7 ⊢ (Base‘(𝑅 ↑_s 𝐾)) = (Base‘(𝑅 ↑_s 𝐾))
18	4	fvexi 6903	. . . . . . . 8 ⊢ 𝐾 ∈ V
19	18	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ V)
20	2, 3, 16, 4	evl1rhm 21843	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s 𝐾)))
21	6, 20	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s 𝐾)))
22	5, 17	rhmf 20256	. . . . . . . . 9 ⊢ (𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s 𝐾)) → 𝑂:𝐵⟶(Base‘(𝑅 ↑_s 𝐾)))
23	21, 22	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑂:𝐵⟶(Base‘(𝑅 ↑_s 𝐾)))
24	12	simpld 496	. . . . . . . 8 ⊢ (𝜑 → (𝑀 · 𝑋) ∈ 𝐵)
25	23, 24	ffvelcdmd 7085	. . . . . . 7 ⊢ (𝜑 → (𝑂‘(𝑀 · 𝑋)) ∈ (Base‘(𝑅 ↑_s 𝐾)))
26	16, 4, 17, 6, 19, 25	pwselbas 17432	. . . . . 6 ⊢ (𝜑 → (𝑂‘(𝑀 · 𝑋)):𝐾⟶𝐾)
27	26	ffnd 6716	. . . . 5 ⊢ (𝜑 → (𝑂‘(𝑀 · 𝑋)) Fn 𝐾)
28		fniniseg 7059	. . . . 5 ⊢ ((𝑂‘(𝑀 · 𝑋)) Fn 𝐾 → (𝑁 ∈ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ↔ (𝑁 ∈ 𝐾 ∧ ((𝑂‘(𝑀 · 𝑋))‘𝑁) = 𝑊)))
29	27, 28	syl 17	. . . 4 ⊢ (𝜑 → (𝑁 ∈ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ↔ (𝑁 ∈ 𝐾 ∧ ((𝑂‘(𝑀 · 𝑋))‘𝑁) = 𝑊)))
30	1, 15, 29	mpbir2and 712	. . 3 ⊢ (𝜑 → 𝑁 ∈ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}))
31		fvex 6902	. . . . . . . 8 ⊢ (𝑂‘(𝑀 · 𝑋)) ∈ V
32	31	cnvex 7913	. . . . . . 7 ⊢ ^◡(𝑂‘(𝑀 · 𝑋)) ∈ V
33	32	imaex 7904	. . . . . 6 ⊢ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ∈ V
34	33	a1i 11	. . . . 5 ⊢ (𝜑 → (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ∈ V)
35		1nn0 12485	. . . . . 6 ⊢ 1 ∈ ℕ₀
36	35	a1i 11	. . . . 5 ⊢ (𝜑 → 1 ∈ ℕ₀)
37		crngring 20062	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
38	6, 37	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ Ring)
39	7, 3, 5	vr1cl 21733	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵)
40	38, 39	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
41		eqid 2733	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
42	41, 5	mgpbas 19988	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘(mulGrp‘𝑃))
43		eqid 2733	. . . . . . . . . . . 12 ⊢ (._g‘(mulGrp‘𝑃)) = (._g‘(mulGrp‘𝑃))
44	42, 43	mulg1 18956	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝐵 → (1(._g‘(mulGrp‘𝑃))𝑋) = 𝑋)
45	40, 44	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (1(._g‘(mulGrp‘𝑃))𝑋) = 𝑋)
46	45	oveq2d 7422	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋)) = (𝑀 · 𝑋))
47		fta1blem.5	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ≠ 𝑊)
48		fta1b.w	. . . . . . . . . . . . 13 ⊢ 𝑊 = (0_g‘𝑅)
49	48, 4, 3, 7, 10, 41, 43	coe1tmfv1 21788	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾 ∧ 1 ∈ ℕ₀) → ((coe₁‘(𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋)))‘1) = 𝑀)
50	38, 9, 36, 49	syl3anc 1372	. . . . . . . . . . 11 ⊢ (𝜑 → ((coe₁‘(𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋)))‘1) = 𝑀)
51		fta1b.z	. . . . . . . . . . . . . . 15 ⊢ 0 = (0_g‘𝑃)
52	3, 51, 48	coe1z 21777	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → (coe₁‘ 0 ) = (ℕ₀ × {𝑊}))
53	38, 52	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (coe₁‘ 0 ) = (ℕ₀ × {𝑊}))
54	53	fveq1d 6891	. . . . . . . . . . . 12 ⊢ (𝜑 → ((coe₁‘ 0 )‘1) = ((ℕ₀ × {𝑊})‘1))
55	48	fvexi 6903	. . . . . . . . . . . . . 14 ⊢ 𝑊 ∈ V
56	55	fvconst2 7202	. . . . . . . . . . . . 13 ⊢ (1 ∈ ℕ₀ → ((ℕ₀ × {𝑊})‘1) = 𝑊)
57	35, 56	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((ℕ₀ × {𝑊})‘1) = 𝑊
58	54, 57	eqtrdi 2789	. . . . . . . . . . 11 ⊢ (𝜑 → ((coe₁‘ 0 )‘1) = 𝑊)
59	47, 50, 58	3netr4d 3019	. . . . . . . . . 10 ⊢ (𝜑 → ((coe₁‘(𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋)))‘1) ≠ ((coe₁‘ 0 )‘1))
60		fveq2 6889	. . . . . . . . . . . 12 ⊢ ((𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋)) = 0 → (coe₁‘(𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋))) = (coe₁‘ 0 ))
61	60	fveq1d 6891	. . . . . . . . . . 11 ⊢ ((𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋)) = 0 → ((coe₁‘(𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋)))‘1) = ((coe₁‘ 0 )‘1))
62	61	necon3i 2974	. . . . . . . . . 10 ⊢ (((coe₁‘(𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋)))‘1) ≠ ((coe₁‘ 0 )‘1) → (𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋)) ≠ 0 )
63	59, 62	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋)) ≠ 0 )
64	46, 63	eqnetrrd 3010	. . . . . . . 8 ⊢ (𝜑 → (𝑀 · 𝑋) ≠ 0 )
65		eldifsn 4790	. . . . . . . 8 ⊢ ((𝑀 · 𝑋) ∈ (𝐵 ∖ { 0 }) ↔ ((𝑀 · 𝑋) ∈ 𝐵 ∧ (𝑀 · 𝑋) ≠ 0 ))
66	24, 64, 65	sylanbrc 584	. . . . . . 7 ⊢ (𝜑 → (𝑀 · 𝑋) ∈ (𝐵 ∖ { 0 }))
67		fta1blem.6	. . . . . . 7 ⊢ (𝜑 → ((𝑀 · 𝑋) ∈ (𝐵 ∖ { 0 }) → (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ≤ (𝐷‘(𝑀 · 𝑋))))
68	66, 67	mpd 15	. . . . . 6 ⊢ (𝜑 → (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ≤ (𝐷‘(𝑀 · 𝑋)))
69	46	fveq2d 6893	. . . . . . 7 ⊢ (𝜑 → (𝐷‘(𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋))) = (𝐷‘(𝑀 · 𝑋)))
70		fta1b.d	. . . . . . . . 9 ⊢ 𝐷 = ( deg₁ ‘𝑅)
71	70, 4, 3, 7, 10, 41, 43, 48	deg1tm 25628	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐾 ∧ 𝑀 ≠ 𝑊) ∧ 1 ∈ ℕ₀) → (𝐷‘(𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋))) = 1)
72	38, 9, 47, 36, 71	syl121anc 1376	. . . . . . 7 ⊢ (𝜑 → (𝐷‘(𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋))) = 1)
73	69, 72	eqtr3d 2775	. . . . . 6 ⊢ (𝜑 → (𝐷‘(𝑀 · 𝑋)) = 1)
74	68, 73	breqtrd 5174	. . . . 5 ⊢ (𝜑 → (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ≤ 1)
75		hashbnd 14293	. . . . 5 ⊢ (((^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ∈ V ∧ 1 ∈ ℕ₀ ∧ (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ≤ 1) → (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ∈ Fin)
76	34, 36, 74, 75	syl3anc 1372	. . . 4 ⊢ (𝜑 → (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ∈ Fin)
77	4, 48	ring0cl 20078	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ 𝐾)
78	38, 77	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ 𝐾)
79		eqid 2733	. . . . . . . . . . . . 13 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
80	3, 79, 4, 5	ply1sclf 21799	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (algSc‘𝑃):𝐾⟶𝐵)
81	38, 80	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (algSc‘𝑃):𝐾⟶𝐵)
82	81, 9	ffvelcdmd 7085	. . . . . . . . . 10 ⊢ (𝜑 → ((algSc‘𝑃)‘𝑀) ∈ 𝐵)
83		eqid 2733	. . . . . . . . . . 11 ⊢ (._r‘𝑃) = (._r‘𝑃)
84		eqid 2733	. . . . . . . . . . 11 ⊢ (._r‘(𝑅 ↑_s 𝐾)) = (._r‘(𝑅 ↑_s 𝐾))
85	5, 83, 84	rhmmul 20257	. . . . . . . . . 10 ⊢ ((𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s 𝐾)) ∧ ((algSc‘𝑃)‘𝑀) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑂‘(((algSc‘𝑃)‘𝑀)(._r‘𝑃)𝑋)) = ((𝑂‘((algSc‘𝑃)‘𝑀))(._r‘(𝑅 ↑_s 𝐾))(𝑂‘𝑋)))
86	21, 82, 40, 85	syl3anc 1372	. . . . . . . . 9 ⊢ (𝜑 → (𝑂‘(((algSc‘𝑃)‘𝑀)(._r‘𝑃)𝑋)) = ((𝑂‘((algSc‘𝑃)‘𝑀))(._r‘(𝑅 ↑_s 𝐾))(𝑂‘𝑋)))
87	3	ply1assa 21715	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
88	6, 87	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ AssAlg)
89	3	ply1sca 21767	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃))
90	6, 89	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
91	90	fveq2d 6893	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
92	4, 91	eqtrid 2785	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝑃)))
93	9, 92	eleqtrd 2836	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ (Base‘(Scalar‘𝑃)))
94		eqid 2733	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
95		eqid 2733	. . . . . . . . . . . 12 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
96	79, 94, 95, 5, 83, 10	asclmul1 21432	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ AssAlg ∧ 𝑀 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑋 ∈ 𝐵) → (((algSc‘𝑃)‘𝑀)(._r‘𝑃)𝑋) = (𝑀 · 𝑋))
97	88, 93, 40, 96	syl3anc 1372	. . . . . . . . . 10 ⊢ (𝜑 → (((algSc‘𝑃)‘𝑀)(._r‘𝑃)𝑋) = (𝑀 · 𝑋))
98	97	fveq2d 6893	. . . . . . . . 9 ⊢ (𝜑 → (𝑂‘(((algSc‘𝑃)‘𝑀)(._r‘𝑃)𝑋)) = (𝑂‘(𝑀 · 𝑋)))
99	23, 82	ffvelcdmd 7085	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂‘((algSc‘𝑃)‘𝑀)) ∈ (Base‘(𝑅 ↑_s 𝐾)))
100	23, 40	ffvelcdmd 7085	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂‘𝑋) ∈ (Base‘(𝑅 ↑_s 𝐾)))
101	16, 17, 6, 19, 99, 100, 11, 84	pwsmulrval 17434	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑂‘((algSc‘𝑃)‘𝑀))(._r‘(𝑅 ↑_s 𝐾))(𝑂‘𝑋)) = ((𝑂‘((algSc‘𝑃)‘𝑀)) ∘_f × (𝑂‘𝑋)))
102	2, 3, 4, 79	evl1sca 21845	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐾) → (𝑂‘((algSc‘𝑃)‘𝑀)) = (𝐾 × {𝑀}))
103	6, 9, 102	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂‘((algSc‘𝑃)‘𝑀)) = (𝐾 × {𝑀}))
104	2, 7, 4	evl1var 21847	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → (𝑂‘𝑋) = ( I ↾ 𝐾))
105	6, 104	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂‘𝑋) = ( I ↾ 𝐾))
106	103, 105	oveq12d 7424	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑂‘((algSc‘𝑃)‘𝑀)) ∘_f × (𝑂‘𝑋)) = ((𝐾 × {𝑀}) ∘_f × ( I ↾ 𝐾)))
107	101, 106	eqtrd 2773	. . . . . . . . 9 ⊢ (𝜑 → ((𝑂‘((algSc‘𝑃)‘𝑀))(._r‘(𝑅 ↑_s 𝐾))(𝑂‘𝑋)) = ((𝐾 × {𝑀}) ∘_f × ( I ↾ 𝐾)))
108	86, 98, 107	3eqtr3d 2781	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘(𝑀 · 𝑋)) = ((𝐾 × {𝑀}) ∘_f × ( I ↾ 𝐾)))
109	108	fveq1d 6891	. . . . . . 7 ⊢ (𝜑 → ((𝑂‘(𝑀 · 𝑋))‘𝑊) = (((𝐾 × {𝑀}) ∘_f × ( I ↾ 𝐾))‘𝑊))
110		fnconstg 6777	. . . . . . . . . 10 ⊢ (𝑀 ∈ 𝐾 → (𝐾 × {𝑀}) Fn 𝐾)
111	9, 110	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 × {𝑀}) Fn 𝐾)
112		fnresi 6677	. . . . . . . . . 10 ⊢ ( I ↾ 𝐾) Fn 𝐾
113	112	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ( I ↾ 𝐾) Fn 𝐾)
114		fnfvof 7684	. . . . . . . . 9 ⊢ ((((𝐾 × {𝑀}) Fn 𝐾 ∧ ( I ↾ 𝐾) Fn 𝐾) ∧ (𝐾 ∈ V ∧ 𝑊 ∈ 𝐾)) → (((𝐾 × {𝑀}) ∘_f × ( I ↾ 𝐾))‘𝑊) = (((𝐾 × {𝑀})‘𝑊) × (( I ↾ 𝐾)‘𝑊)))
115	111, 113, 19, 78, 114	syl22anc 838	. . . . . . . 8 ⊢ (𝜑 → (((𝐾 × {𝑀}) ∘_f × ( I ↾ 𝐾))‘𝑊) = (((𝐾 × {𝑀})‘𝑊) × (( I ↾ 𝐾)‘𝑊)))
116		fvconst2g 7200	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝐾 ∧ 𝑊 ∈ 𝐾) → ((𝐾 × {𝑀})‘𝑊) = 𝑀)
117	9, 78, 116	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾 × {𝑀})‘𝑊) = 𝑀)
118		fvresi 7168	. . . . . . . . . . 11 ⊢ (𝑊 ∈ 𝐾 → (( I ↾ 𝐾)‘𝑊) = 𝑊)
119	78, 118	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (( I ↾ 𝐾)‘𝑊) = 𝑊)
120	117, 119	oveq12d 7424	. . . . . . . . 9 ⊢ (𝜑 → (((𝐾 × {𝑀})‘𝑊) × (( I ↾ 𝐾)‘𝑊)) = (𝑀 × 𝑊))
121	4, 11, 48	ringrz 20102	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝑀 × 𝑊) = 𝑊)
122	38, 9, 121	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 × 𝑊) = 𝑊)
123	120, 122	eqtrd 2773	. . . . . . . 8 ⊢ (𝜑 → (((𝐾 × {𝑀})‘𝑊) × (( I ↾ 𝐾)‘𝑊)) = 𝑊)
124	115, 123	eqtrd 2773	. . . . . . 7 ⊢ (𝜑 → (((𝐾 × {𝑀}) ∘_f × ( I ↾ 𝐾))‘𝑊) = 𝑊)
125	109, 124	eqtrd 2773	. . . . . 6 ⊢ (𝜑 → ((𝑂‘(𝑀 · 𝑋))‘𝑊) = 𝑊)
126		fniniseg 7059	. . . . . . 7 ⊢ ((𝑂‘(𝑀 · 𝑋)) Fn 𝐾 → (𝑊 ∈ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ↔ (𝑊 ∈ 𝐾 ∧ ((𝑂‘(𝑀 · 𝑋))‘𝑊) = 𝑊)))
127	27, 126	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑊 ∈ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ↔ (𝑊 ∈ 𝐾 ∧ ((𝑂‘(𝑀 · 𝑋))‘𝑊) = 𝑊)))
128	78, 125, 127	mpbir2and 712	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}))
129	128	snssd 4812	. . . 4 ⊢ (𝜑 → {𝑊} ⊆ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}))
130		hashsng 14326	. . . . . . 7 ⊢ (𝑊 ∈ 𝐾 → (♯‘{𝑊}) = 1)
131	78, 130	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘{𝑊}) = 1)
132		ssdomg 8993	. . . . . . . . . 10 ⊢ ((^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ∈ V → ({𝑊} ⊆ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) → {𝑊} ≼ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})))
133	33, 129, 132	mpsyl 68	. . . . . . . . 9 ⊢ (𝜑 → {𝑊} ≼ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}))
134		snfi 9041	. . . . . . . . . 10 ⊢ {𝑊} ∈ Fin
135		hashdom 14336	. . . . . . . . . 10 ⊢ (({𝑊} ∈ Fin ∧ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ∈ V) → ((♯‘{𝑊}) ≤ (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ↔ {𝑊} ≼ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})))
136	134, 33, 135	mp2an 691	. . . . . . . . 9 ⊢ ((♯‘{𝑊}) ≤ (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ↔ {𝑊} ≼ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}))
137	133, 136	sylibr 233	. . . . . . . 8 ⊢ (𝜑 → (♯‘{𝑊}) ≤ (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})))
138	131, 137	eqbrtrrd 5172	. . . . . . 7 ⊢ (𝜑 → 1 ≤ (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})))
139		hashcl 14313	. . . . . . . . . 10 ⊢ ((^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ∈ Fin → (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ∈ ℕ₀)
140	76, 139	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ∈ ℕ₀)
141	140	nn0red 12530	. . . . . . . 8 ⊢ (𝜑 → (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ∈ ℝ)
142		1re 11211	. . . . . . . 8 ⊢ 1 ∈ ℝ
143		letri3 11296	. . . . . . . 8 ⊢ (((♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ∈ ℝ ∧ 1 ∈ ℝ) → ((♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) = 1 ↔ ((♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ≤ 1 ∧ 1 ≤ (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})))))
144	141, 142, 143	sylancl 587	. . . . . . 7 ⊢ (𝜑 → ((♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) = 1 ↔ ((♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ≤ 1 ∧ 1 ≤ (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})))))
145	74, 138, 144	mpbir2and 712	. . . . . 6 ⊢ (𝜑 → (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) = 1)
146	131, 145	eqtr4d 2776	. . . . 5 ⊢ (𝜑 → (♯‘{𝑊}) = (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})))
147		hashen 14304	. . . . . 6 ⊢ (({𝑊} ∈ Fin ∧ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ∈ Fin) → ((♯‘{𝑊}) = (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ↔ {𝑊} ≈ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})))
148	134, 76, 147	sylancr 588	. . . . 5 ⊢ (𝜑 → ((♯‘{𝑊}) = (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ↔ {𝑊} ≈ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})))
149	146, 148	mpbid 231	. . . 4 ⊢ (𝜑 → {𝑊} ≈ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}))
150		fisseneq 9254	. . . 4 ⊢ (((^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ∈ Fin ∧ {𝑊} ⊆ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ∧ {𝑊} ≈ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) → {𝑊} = (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}))
151	76, 129, 149, 150	syl3anc 1372	. . 3 ⊢ (𝜑 → {𝑊} = (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}))
152	30, 151	eleqtrrd 2837	. 2 ⊢ (𝜑 → 𝑁 ∈ {𝑊})
153		elsni 4645	. 2 ⊢ (𝑁 ∈ {𝑊} → 𝑁 = 𝑊)
154	152, 153	syl 17	1 ⊢ (𝜑 → 𝑁 = 𝑊)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 Vcvv 3475 ∖ cdif 3945 ⊆ wss 3948 {csn 4628 class class class wbr 5148 I cid 5573 × cxp 5674 ^◡ccnv 5675 ↾ cres 5678 “ cima 5679 Fn wfn 6536 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 ∘_f cof 7665 ≈ cen 8933 ≼ cdom 8934 Fincfn 8936 ℝcr 11106 1c1 11108 ≤ cle 11246 ℕ₀cn0 12469 ♯chash 14287 Basecbs 17141 ._rcmulr 17195 Scalarcsca 17197 ·_𝑠 cvsca 17198 0_gc0g 17382 ↑_s cpws 17389 ._gcmg 18945 mulGrpcmgp 19982 Ringcrg 20050 CRingccrg 20051 RingHom crh 20241 AssAlgcasa 21397 algSccascl 21399 var₁cv1 21692 Poly₁cpl1 21693 coe₁cco1 21694 eval₁ce1 21825 deg₁ cdg1 25561

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 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-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 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-se 5632 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 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-ofr 7668 df-om 7853 df-1st 7972 df-2nd 7973 df-supp 8144 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-oadd 8467 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-sup 9434 df-oi 9502 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-xnn0 12542 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-fzo 13625 df-seq 13964 df-hash 14288 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-starv 17209 df-sca 17210 df-vsca 17211 df-ip 17212 df-tset 17213 df-ple 17214 df-ds 17216 df-unif 17217 df-hom 17218 df-cco 17219 df-0g 17384 df-gsum 17385 df-prds 17390 df-pws 17392 df-mre 17527 df-mrc 17528 df-acs 17530 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-mhm 18668 df-submnd 18669 df-grp 18819 df-minusg 18820 df-sbg 18821 df-mulg 18946 df-subg 18998 df-ghm 19085 df-cntz 19176 df-cmn 19645 df-abl 19646 df-mgp 19983 df-ur 20000 df-srg 20004 df-ring 20052 df-cring 20053 df-rnghom 20244 df-subrg 20354 df-lmod 20466 df-lss 20536 df-lsp 20576 df-cnfld 20938 df-assa 21400 df-asp 21401 df-ascl 21402 df-psr 21454 df-mvr 21455 df-mpl 21456 df-opsr 21458 df-evls 21627 df-evl 21628 df-psr1 21696 df-vr1 21697 df-ply1 21698 df-coe1 21699 df-evl1 21827 df-mdeg 25562 df-deg1 25563

This theorem is referenced by: fta1b 25679

W3C validator