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Theorem fta1blem 25921

Description: Lemma for fta1b 25922. (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 22076	. . . . . . 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 22083	. . . . . 6 ⊢ (𝜑 → ((𝑀 · 𝑋) ∈ 𝐵 ∧ ((𝑂‘(𝑀 · 𝑋))‘𝑁) = (𝑀 × 𝑁)))
13	12	simprd 494	. . . . 5 ⊢ (𝜑 → ((𝑂‘(𝑀 · 𝑋))‘𝑁) = (𝑀 × 𝑁))
14		fta1blem.4	. . . . 5 ⊢ (𝜑 → (𝑀 × 𝑁) = 𝑊)
15	13, 14	eqtrd 2770	. . . 4 ⊢ (𝜑 → ((𝑂‘(𝑀 · 𝑋))‘𝑁) = 𝑊)
16		eqid 2730	. . . . . . 7 ⊢ (𝑅 ↑_s 𝐾) = (𝑅 ↑_s 𝐾)
17		eqid 2730	. . . . . . 7 ⊢ (Base‘(𝑅 ↑_s 𝐾)) = (Base‘(𝑅 ↑_s 𝐾))
18	4	fvexi 6904	. . . . . . . 8 ⊢ 𝐾 ∈ V
19	18	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ V)
20	2, 3, 16, 4	evl1rhm 22071	. . . . . . . . . 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 20376	. . . . . . . . 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 493	. . . . . . . 8 ⊢ (𝜑 → (𝑀 · 𝑋) ∈ 𝐵)
25	23, 24	ffvelcdmd 7086	. . . . . . 7 ⊢ (𝜑 → (𝑂‘(𝑀 · 𝑋)) ∈ (Base‘(𝑅 ↑_s 𝐾)))
26	16, 4, 17, 6, 19, 25	pwselbas 17439	. . . . . 6 ⊢ (𝜑 → (𝑂‘(𝑀 · 𝑋)):𝐾⟶𝐾)
27	26	ffnd 6717	. . . . 5 ⊢ (𝜑 → (𝑂‘(𝑀 · 𝑋)) Fn 𝐾)
28		fniniseg 7060	. . . . 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 709	. . 3 ⊢ (𝜑 → 𝑁 ∈ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}))
31		fvex 6903	. . . . . . . 8 ⊢ (𝑂‘(𝑀 · 𝑋)) ∈ V
32	31	cnvex 7918	. . . . . . 7 ⊢ ^◡(𝑂‘(𝑀 · 𝑋)) ∈ V
33	32	imaex 7909	. . . . . 6 ⊢ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ∈ V
34	33	a1i 11	. . . . 5 ⊢ (𝜑 → (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ∈ V)
35		1nn0 12492	. . . . . 6 ⊢ 1 ∈ ℕ₀
36	35	a1i 11	. . . . 5 ⊢ (𝜑 → 1 ∈ ℕ₀)
37		crngring 20139	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
38	6, 37	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ Ring)
39	7, 3, 5	vr1cl 21960	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵)
40	38, 39	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
41		eqid 2730	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
42	41, 5	mgpbas 20034	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘(mulGrp‘𝑃))
43		eqid 2730	. . . . . . . . . . . 12 ⊢ (._g‘(mulGrp‘𝑃)) = (._g‘(mulGrp‘𝑃))
44	42, 43	mulg1 18997	. . . . . . . . . . 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 7427	. . . . . . . . 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 22016	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾 ∧ 1 ∈ ℕ₀) → ((coe₁‘(𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋)))‘1) = 𝑀)
50	38, 9, 36, 49	syl3anc 1369	. . . . . . . . . . 11 ⊢ (𝜑 → ((coe₁‘(𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋)))‘1) = 𝑀)
51		fta1b.z	. . . . . . . . . . . . . . 15 ⊢ 0 = (0_g‘𝑃)
52	3, 51, 48	coe1z 22005	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → (coe₁‘ 0 ) = (ℕ₀ × {𝑊}))
53	38, 52	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (coe₁‘ 0 ) = (ℕ₀ × {𝑊}))
54	53	fveq1d 6892	. . . . . . . . . . . 12 ⊢ (𝜑 → ((coe₁‘ 0 )‘1) = ((ℕ₀ × {𝑊})‘1))
55	48	fvexi 6904	. . . . . . . . . . . . . 14 ⊢ 𝑊 ∈ V
56	55	fvconst2 7206	. . . . . . . . . . . . 13 ⊢ (1 ∈ ℕ₀ → ((ℕ₀ × {𝑊})‘1) = 𝑊)
57	35, 56	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((ℕ₀ × {𝑊})‘1) = 𝑊
58	54, 57	eqtrdi 2786	. . . . . . . . . . 11 ⊢ (𝜑 → ((coe₁‘ 0 )‘1) = 𝑊)
59	47, 50, 58	3netr4d 3016	. . . . . . . . . 10 ⊢ (𝜑 → ((coe₁‘(𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋)))‘1) ≠ ((coe₁‘ 0 )‘1))
60		fveq2 6890	. . . . . . . . . . . 12 ⊢ ((𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋)) = 0 → (coe₁‘(𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋))) = (coe₁‘ 0 ))
61	60	fveq1d 6892	. . . . . . . . . . 11 ⊢ ((𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋)) = 0 → ((coe₁‘(𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋)))‘1) = ((coe₁‘ 0 )‘1))
62	61	necon3i 2971	. . . . . . . . . 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 3007	. . . . . . . 8 ⊢ (𝜑 → (𝑀 · 𝑋) ≠ 0 )
65		eldifsn 4789	. . . . . . . 8 ⊢ ((𝑀 · 𝑋) ∈ (𝐵 ∖ { 0 }) ↔ ((𝑀 · 𝑋) ∈ 𝐵 ∧ (𝑀 · 𝑋) ≠ 0 ))
66	24, 64, 65	sylanbrc 581	. . . . . . 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 6894	. . . . . . 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 25871	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐾 ∧ 𝑀 ≠ 𝑊) ∧ 1 ∈ ℕ₀) → (𝐷‘(𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋))) = 1)
72	38, 9, 47, 36, 71	syl121anc 1373	. . . . . . 7 ⊢ (𝜑 → (𝐷‘(𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋))) = 1)
73	69, 72	eqtr3d 2772	. . . . . 6 ⊢ (𝜑 → (𝐷‘(𝑀 · 𝑋)) = 1)
74	68, 73	breqtrd 5173	. . . . 5 ⊢ (𝜑 → (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ≤ 1)
75		hashbnd 14300	. . . . 5 ⊢ (((^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ∈ V ∧ 1 ∈ ℕ₀ ∧ (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ≤ 1) → (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ∈ Fin)
76	34, 36, 74, 75	syl3anc 1369	. . . 4 ⊢ (𝜑 → (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ∈ Fin)
77	4, 48	ring0cl 20155	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ 𝐾)
78	38, 77	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ 𝐾)
79		eqid 2730	. . . . . . . . . . . . 13 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
80	3, 79, 4, 5	ply1sclf 22027	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (algSc‘𝑃):𝐾⟶𝐵)
81	38, 80	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (algSc‘𝑃):𝐾⟶𝐵)
82	81, 9	ffvelcdmd 7086	. . . . . . . . . 10 ⊢ (𝜑 → ((algSc‘𝑃)‘𝑀) ∈ 𝐵)
83		eqid 2730	. . . . . . . . . . 11 ⊢ (._r‘𝑃) = (._r‘𝑃)
84		eqid 2730	. . . . . . . . . . 11 ⊢ (._r‘(𝑅 ↑_s 𝐾)) = (._r‘(𝑅 ↑_s 𝐾))
85	5, 83, 84	rhmmul 20377	. . . . . . . . . 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 1369	. . . . . . . . 9 ⊢ (𝜑 → (𝑂‘(((algSc‘𝑃)‘𝑀)(._r‘𝑃)𝑋)) = ((𝑂‘((algSc‘𝑃)‘𝑀))(._r‘(𝑅 ↑_s 𝐾))(𝑂‘𝑋)))
87	3	ply1assa 21942	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
88	6, 87	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ AssAlg)
89	3	ply1sca 21995	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃))
90	6, 89	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
91	90	fveq2d 6894	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
92	4, 91	eqtrid 2782	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝑃)))
93	9, 92	eleqtrd 2833	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ (Base‘(Scalar‘𝑃)))
94		eqid 2730	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
95		eqid 2730	. . . . . . . . . . . 12 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
96	79, 94, 95, 5, 83, 10	asclmul1 21659	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ AssAlg ∧ 𝑀 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑋 ∈ 𝐵) → (((algSc‘𝑃)‘𝑀)(._r‘𝑃)𝑋) = (𝑀 · 𝑋))
97	88, 93, 40, 96	syl3anc 1369	. . . . . . . . . 10 ⊢ (𝜑 → (((algSc‘𝑃)‘𝑀)(._r‘𝑃)𝑋) = (𝑀 · 𝑋))
98	97	fveq2d 6894	. . . . . . . . 9 ⊢ (𝜑 → (𝑂‘(((algSc‘𝑃)‘𝑀)(._r‘𝑃)𝑋)) = (𝑂‘(𝑀 · 𝑋)))
99	23, 82	ffvelcdmd 7086	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂‘((algSc‘𝑃)‘𝑀)) ∈ (Base‘(𝑅 ↑_s 𝐾)))
100	23, 40	ffvelcdmd 7086	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂‘𝑋) ∈ (Base‘(𝑅 ↑_s 𝐾)))
101	16, 17, 6, 19, 99, 100, 11, 84	pwsmulrval 17441	. . . . . . . . . 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 22073	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐾) → (𝑂‘((algSc‘𝑃)‘𝑀)) = (𝐾 × {𝑀}))
103	6, 9, 102	syl2anc 582	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂‘((algSc‘𝑃)‘𝑀)) = (𝐾 × {𝑀}))
104	2, 7, 4	evl1var 22075	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → (𝑂‘𝑋) = ( I ↾ 𝐾))
105	6, 104	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂‘𝑋) = ( I ↾ 𝐾))
106	103, 105	oveq12d 7429	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑂‘((algSc‘𝑃)‘𝑀)) ∘_f × (𝑂‘𝑋)) = ((𝐾 × {𝑀}) ∘_f × ( I ↾ 𝐾)))
107	101, 106	eqtrd 2770	. . . . . . . . 9 ⊢ (𝜑 → ((𝑂‘((algSc‘𝑃)‘𝑀))(._r‘(𝑅 ↑_s 𝐾))(𝑂‘𝑋)) = ((𝐾 × {𝑀}) ∘_f × ( I ↾ 𝐾)))
108	86, 98, 107	3eqtr3d 2778	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘(𝑀 · 𝑋)) = ((𝐾 × {𝑀}) ∘_f × ( I ↾ 𝐾)))
109	108	fveq1d 6892	. . . . . . 7 ⊢ (𝜑 → ((𝑂‘(𝑀 · 𝑋))‘𝑊) = (((𝐾 × {𝑀}) ∘_f × ( I ↾ 𝐾))‘𝑊))
110		fnconstg 6778	. . . . . . . . . 10 ⊢ (𝑀 ∈ 𝐾 → (𝐾 × {𝑀}) Fn 𝐾)
111	9, 110	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 × {𝑀}) Fn 𝐾)
112		fnresi 6678	. . . . . . . . . 10 ⊢ ( I ↾ 𝐾) Fn 𝐾
113	112	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ( I ↾ 𝐾) Fn 𝐾)
114		fnfvof 7689	. . . . . . . . 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 835	. . . . . . . 8 ⊢ (𝜑 → (((𝐾 × {𝑀}) ∘_f × ( I ↾ 𝐾))‘𝑊) = (((𝐾 × {𝑀})‘𝑊) × (( I ↾ 𝐾)‘𝑊)))
116		fvconst2g 7204	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝐾 ∧ 𝑊 ∈ 𝐾) → ((𝐾 × {𝑀})‘𝑊) = 𝑀)
117	9, 78, 116	syl2anc 582	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾 × {𝑀})‘𝑊) = 𝑀)
118		fvresi 7172	. . . . . . . . . . 11 ⊢ (𝑊 ∈ 𝐾 → (( I ↾ 𝐾)‘𝑊) = 𝑊)
119	78, 118	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (( I ↾ 𝐾)‘𝑊) = 𝑊)
120	117, 119	oveq12d 7429	. . . . . . . . 9 ⊢ (𝜑 → (((𝐾 × {𝑀})‘𝑊) × (( I ↾ 𝐾)‘𝑊)) = (𝑀 × 𝑊))
121	4, 11, 48	ringrz 20182	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝑀 × 𝑊) = 𝑊)
122	38, 9, 121	syl2anc 582	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 × 𝑊) = 𝑊)
123	120, 122	eqtrd 2770	. . . . . . . 8 ⊢ (𝜑 → (((𝐾 × {𝑀})‘𝑊) × (( I ↾ 𝐾)‘𝑊)) = 𝑊)
124	115, 123	eqtrd 2770	. . . . . . 7 ⊢ (𝜑 → (((𝐾 × {𝑀}) ∘_f × ( I ↾ 𝐾))‘𝑊) = 𝑊)
125	109, 124	eqtrd 2770	. . . . . 6 ⊢ (𝜑 → ((𝑂‘(𝑀 · 𝑋))‘𝑊) = 𝑊)
126		fniniseg 7060	. . . . . . 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 709	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}))
129	128	snssd 4811	. . . 4 ⊢ (𝜑 → {𝑊} ⊆ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}))
130		hashsng 14333	. . . . . . 7 ⊢ (𝑊 ∈ 𝐾 → (♯‘{𝑊}) = 1)
131	78, 130	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘{𝑊}) = 1)
132		ssdomg 8998	. . . . . . . . . 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 9046	. . . . . . . . . 10 ⊢ {𝑊} ∈ Fin
135		hashdom 14343	. . . . . . . . . 10 ⊢ (({𝑊} ∈ Fin ∧ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ∈ V) → ((♯‘{𝑊}) ≤ (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ↔ {𝑊} ≼ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})))
136	134, 33, 135	mp2an 688	. . . . . . . . 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 5171	. . . . . . 7 ⊢ (𝜑 → 1 ≤ (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})))
139		hashcl 14320	. . . . . . . . . 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 12537	. . . . . . . 8 ⊢ (𝜑 → (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ∈ ℝ)
142		1re 11218	. . . . . . . 8 ⊢ 1 ∈ ℝ
143		letri3 11303	. . . . . . . 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 584	. . . . . . 7 ⊢ (𝜑 → ((♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) = 1 ↔ ((♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ≤ 1 ∧ 1 ≤ (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})))))
145	74, 138, 144	mpbir2and 709	. . . . . 6 ⊢ (𝜑 → (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) = 1)
146	131, 145	eqtr4d 2773	. . . . 5 ⊢ (𝜑 → (♯‘{𝑊}) = (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})))
147		hashen 14311	. . . . . 6 ⊢ (({𝑊} ∈ Fin ∧ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ∈ Fin) → ((♯‘{𝑊}) = (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ↔ {𝑊} ≈ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})))
148	134, 76, 147	sylancr 585	. . . . 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 9259	. . . 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 1369	. . 3 ⊢ (𝜑 → {𝑊} = (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}))
152	30, 151	eleqtrrd 2834	. 2 ⊢ (𝜑 → 𝑁 ∈ {𝑊})
153		elsni 4644	. 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 394 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 Vcvv 3472 ∖ cdif 3944 ⊆ wss 3947 {csn 4627 class class class wbr 5147 I cid 5572 × cxp 5673 ^◡ccnv 5674 ↾ cres 5677 “ cima 5678 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ∘_f cof 7670 ≈ cen 8938 ≼ cdom 8939 Fincfn 8941 ℝcr 11111 1c1 11113 ≤ cle 11253 ℕ₀cn0 12476 ♯chash 14294 Basecbs 17148 ._rcmulr 17202 Scalarcsca 17204 ·_𝑠 cvsca 17205 0_gc0g 17389 ↑_s cpws 17396 ._gcmg 18986 mulGrpcmgp 20028 Ringcrg 20127 CRingccrg 20128 RingHom crh 20360 AssAlgcasa 21624 algSccascl 21626 var₁cv1 21919 Poly₁cpl1 21920 coe₁cco1 21921 eval₁ce1 22053 deg₁ cdg1 25804

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 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

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-rmo 3374 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-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 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-se 5631 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-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-ofr 7673 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-oadd 8472 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-oi 9507 df-card 9936 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-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13489 df-fzo 13632 df-seq 13971 df-hash 14295 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-0g 17391 df-gsum 17392 df-prds 17397 df-pws 17399 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-mulg 18987 df-subg 19039 df-ghm 19128 df-cntz 19222 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-srg 20081 df-ring 20129 df-cring 20130 df-rhm 20363 df-subrng 20434 df-subrg 20459 df-lmod 20616 df-lss 20687 df-lsp 20727 df-cnfld 21145 df-assa 21627 df-asp 21628 df-ascl 21629 df-psr 21681 df-mvr 21682 df-mpl 21683 df-opsr 21685 df-evls 21854 df-evl 21855 df-psr1 21923 df-vr1 21924 df-ply1 21925 df-coe1 21926 df-evl1 22055 df-mdeg 25805 df-deg1 25806

This theorem is referenced by: fta1b 25922

W3C validator