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Theorem fta1blem 26133

Description: Lemma for fta1b 26134. (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 22275	. . . . . . 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 22282	. . . . . 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 2768	. . . 4 ⊢ (𝜑 → ((𝑂‘(𝑀 · 𝑋))‘𝑁) = 𝑊)
16		eqid 2728	. . . . . . 7 ⊢ (𝑅 ↑_s 𝐾) = (𝑅 ↑_s 𝐾)
17		eqid 2728	. . . . . . 7 ⊢ (Base‘(𝑅 ↑_s 𝐾)) = (Base‘(𝑅 ↑_s 𝐾))
18	4	fvexi 6916	. . . . . . . 8 ⊢ 𝐾 ∈ V
19	18	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ V)
20	2, 3, 16, 4	evl1rhm 22270	. . . . . . . . . 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 20438	. . . . . . . . 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 7100	. . . . . . 7 ⊢ (𝜑 → (𝑂‘(𝑀 · 𝑋)) ∈ (Base‘(𝑅 ↑_s 𝐾)))
26	16, 4, 17, 6, 19, 25	pwselbas 17480	. . . . . 6 ⊢ (𝜑 → (𝑂‘(𝑀 · 𝑋)):𝐾⟶𝐾)
27	26	ffnd 6728	. . . . 5 ⊢ (𝜑 → (𝑂‘(𝑀 · 𝑋)) Fn 𝐾)
28		fniniseg 7074	. . . . 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 711	. . 3 ⊢ (𝜑 → 𝑁 ∈ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}))
31		fvex 6915	. . . . . . . 8 ⊢ (𝑂‘(𝑀 · 𝑋)) ∈ V
32	31	cnvex 7941	. . . . . . 7 ⊢ ^◡(𝑂‘(𝑀 · 𝑋)) ∈ V
33	32	imaex 7930	. . . . . 6 ⊢ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ∈ V
34	33	a1i 11	. . . . 5 ⊢ (𝜑 → (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ∈ V)
35		1nn0 12528	. . . . . 6 ⊢ 1 ∈ ℕ₀
36	35	a1i 11	. . . . 5 ⊢ (𝜑 → 1 ∈ ℕ₀)
37		crngring 20199	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
38	6, 37	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ Ring)
39	7, 3, 5	vr1cl 22155	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵)
40	38, 39	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
41		eqid 2728	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
42	41, 5	mgpbas 20094	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘(mulGrp‘𝑃))
43		eqid 2728	. . . . . . . . . . . 12 ⊢ (._g‘(mulGrp‘𝑃)) = (._g‘(mulGrp‘𝑃))
44	42, 43	mulg1 19050	. . . . . . . . . . 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 7442	. . . . . . . . 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 22212	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾 ∧ 1 ∈ ℕ₀) → ((coe₁‘(𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋)))‘1) = 𝑀)
50	38, 9, 36, 49	syl3anc 1368	. . . . . . . . . . 11 ⊢ (𝜑 → ((coe₁‘(𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋)))‘1) = 𝑀)
51		fta1b.z	. . . . . . . . . . . . . . 15 ⊢ 0 = (0_g‘𝑃)
52	3, 51, 48	coe1z 22201	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → (coe₁‘ 0 ) = (ℕ₀ × {𝑊}))
53	38, 52	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (coe₁‘ 0 ) = (ℕ₀ × {𝑊}))
54	53	fveq1d 6904	. . . . . . . . . . . 12 ⊢ (𝜑 → ((coe₁‘ 0 )‘1) = ((ℕ₀ × {𝑊})‘1))
55	48	fvexi 6916	. . . . . . . . . . . . . 14 ⊢ 𝑊 ∈ V
56	55	fvconst2 7222	. . . . . . . . . . . . 13 ⊢ (1 ∈ ℕ₀ → ((ℕ₀ × {𝑊})‘1) = 𝑊)
57	35, 56	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((ℕ₀ × {𝑊})‘1) = 𝑊
58	54, 57	eqtrdi 2784	. . . . . . . . . . 11 ⊢ (𝜑 → ((coe₁‘ 0 )‘1) = 𝑊)
59	47, 50, 58	3netr4d 3015	. . . . . . . . . 10 ⊢ (𝜑 → ((coe₁‘(𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋)))‘1) ≠ ((coe₁‘ 0 )‘1))
60		fveq2 6902	. . . . . . . . . . . 12 ⊢ ((𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋)) = 0 → (coe₁‘(𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋))) = (coe₁‘ 0 ))
61	60	fveq1d 6904	. . . . . . . . . . 11 ⊢ ((𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋)) = 0 → ((coe₁‘(𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋)))‘1) = ((coe₁‘ 0 )‘1))
62	61	necon3i 2970	. . . . . . . . . 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 3006	. . . . . . . 8 ⊢ (𝜑 → (𝑀 · 𝑋) ≠ 0 )
65		eldifsn 4795	. . . . . . . 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 6906	. . . . . . 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 26082	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐾 ∧ 𝑀 ≠ 𝑊) ∧ 1 ∈ ℕ₀) → (𝐷‘(𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋))) = 1)
72	38, 9, 47, 36, 71	syl121anc 1372	. . . . . . 7 ⊢ (𝜑 → (𝐷‘(𝑀 · (1(._g‘(mulGrp‘𝑃))𝑋))) = 1)
73	69, 72	eqtr3d 2770	. . . . . 6 ⊢ (𝜑 → (𝐷‘(𝑀 · 𝑋)) = 1)
74	68, 73	breqtrd 5178	. . . . 5 ⊢ (𝜑 → (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ≤ 1)
75		hashbnd 14337	. . . . 5 ⊢ (((^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ∈ V ∧ 1 ∈ ℕ₀ ∧ (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ≤ 1) → (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ∈ Fin)
76	34, 36, 74, 75	syl3anc 1368	. . . 4 ⊢ (𝜑 → (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ∈ Fin)
77	4, 48	ring0cl 20217	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ 𝐾)
78	38, 77	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ 𝐾)
79		eqid 2728	. . . . . . . . . . . . 13 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
80	3, 79, 4, 5	ply1sclf 22223	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (algSc‘𝑃):𝐾⟶𝐵)
81	38, 80	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (algSc‘𝑃):𝐾⟶𝐵)
82	81, 9	ffvelcdmd 7100	. . . . . . . . . 10 ⊢ (𝜑 → ((algSc‘𝑃)‘𝑀) ∈ 𝐵)
83		eqid 2728	. . . . . . . . . . 11 ⊢ (._r‘𝑃) = (._r‘𝑃)
84		eqid 2728	. . . . . . . . . . 11 ⊢ (._r‘(𝑅 ↑_s 𝐾)) = (._r‘(𝑅 ↑_s 𝐾))
85	5, 83, 84	rhmmul 20439	. . . . . . . . . 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 1368	. . . . . . . . 9 ⊢ (𝜑 → (𝑂‘(((algSc‘𝑃)‘𝑀)(._r‘𝑃)𝑋)) = ((𝑂‘((algSc‘𝑃)‘𝑀))(._r‘(𝑅 ↑_s 𝐾))(𝑂‘𝑋)))
87	3	ply1assa 22137	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
88	6, 87	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ AssAlg)
89	3	ply1sca 22190	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃))
90	6, 89	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
91	90	fveq2d 6906	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
92	4, 91	eqtrid 2780	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝑃)))
93	9, 92	eleqtrd 2831	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ (Base‘(Scalar‘𝑃)))
94		eqid 2728	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
95		eqid 2728	. . . . . . . . . . . 12 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
96	79, 94, 95, 5, 83, 10	asclmul1 21833	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ AssAlg ∧ 𝑀 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑋 ∈ 𝐵) → (((algSc‘𝑃)‘𝑀)(._r‘𝑃)𝑋) = (𝑀 · 𝑋))
97	88, 93, 40, 96	syl3anc 1368	. . . . . . . . . 10 ⊢ (𝜑 → (((algSc‘𝑃)‘𝑀)(._r‘𝑃)𝑋) = (𝑀 · 𝑋))
98	97	fveq2d 6906	. . . . . . . . 9 ⊢ (𝜑 → (𝑂‘(((algSc‘𝑃)‘𝑀)(._r‘𝑃)𝑋)) = (𝑂‘(𝑀 · 𝑋)))
99	23, 82	ffvelcdmd 7100	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂‘((algSc‘𝑃)‘𝑀)) ∈ (Base‘(𝑅 ↑_s 𝐾)))
100	23, 40	ffvelcdmd 7100	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂‘𝑋) ∈ (Base‘(𝑅 ↑_s 𝐾)))
101	16, 17, 6, 19, 99, 100, 11, 84	pwsmulrval 17482	. . . . . . . . . 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 22272	. . . . . . . . . . . 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 22274	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → (𝑂‘𝑋) = ( I ↾ 𝐾))
105	6, 104	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂‘𝑋) = ( I ↾ 𝐾))
106	103, 105	oveq12d 7444	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑂‘((algSc‘𝑃)‘𝑀)) ∘_f × (𝑂‘𝑋)) = ((𝐾 × {𝑀}) ∘_f × ( I ↾ 𝐾)))
107	101, 106	eqtrd 2768	. . . . . . . . 9 ⊢ (𝜑 → ((𝑂‘((algSc‘𝑃)‘𝑀))(._r‘(𝑅 ↑_s 𝐾))(𝑂‘𝑋)) = ((𝐾 × {𝑀}) ∘_f × ( I ↾ 𝐾)))
108	86, 98, 107	3eqtr3d 2776	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘(𝑀 · 𝑋)) = ((𝐾 × {𝑀}) ∘_f × ( I ↾ 𝐾)))
109	108	fveq1d 6904	. . . . . . 7 ⊢ (𝜑 → ((𝑂‘(𝑀 · 𝑋))‘𝑊) = (((𝐾 × {𝑀}) ∘_f × ( I ↾ 𝐾))‘𝑊))
110		fnconstg 6790	. . . . . . . . . 10 ⊢ (𝑀 ∈ 𝐾 → (𝐾 × {𝑀}) Fn 𝐾)
111	9, 110	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 × {𝑀}) Fn 𝐾)
112		fnresi 6689	. . . . . . . . . 10 ⊢ ( I ↾ 𝐾) Fn 𝐾
113	112	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ( I ↾ 𝐾) Fn 𝐾)
114		fnfvof 7709	. . . . . . . . 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 837	. . . . . . . 8 ⊢ (𝜑 → (((𝐾 × {𝑀}) ∘_f × ( I ↾ 𝐾))‘𝑊) = (((𝐾 × {𝑀})‘𝑊) × (( I ↾ 𝐾)‘𝑊)))
116		fvconst2g 7220	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝐾 ∧ 𝑊 ∈ 𝐾) → ((𝐾 × {𝑀})‘𝑊) = 𝑀)
117	9, 78, 116	syl2anc 582	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾 × {𝑀})‘𝑊) = 𝑀)
118		fvresi 7188	. . . . . . . . . . 11 ⊢ (𝑊 ∈ 𝐾 → (( I ↾ 𝐾)‘𝑊) = 𝑊)
119	78, 118	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (( I ↾ 𝐾)‘𝑊) = 𝑊)
120	117, 119	oveq12d 7444	. . . . . . . . 9 ⊢ (𝜑 → (((𝐾 × {𝑀})‘𝑊) × (( I ↾ 𝐾)‘𝑊)) = (𝑀 × 𝑊))
121	4, 11, 48	ringrz 20244	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝑀 × 𝑊) = 𝑊)
122	38, 9, 121	syl2anc 582	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 × 𝑊) = 𝑊)
123	120, 122	eqtrd 2768	. . . . . . . 8 ⊢ (𝜑 → (((𝐾 × {𝑀})‘𝑊) × (( I ↾ 𝐾)‘𝑊)) = 𝑊)
124	115, 123	eqtrd 2768	. . . . . . 7 ⊢ (𝜑 → (((𝐾 × {𝑀}) ∘_f × ( I ↾ 𝐾))‘𝑊) = 𝑊)
125	109, 124	eqtrd 2768	. . . . . 6 ⊢ (𝜑 → ((𝑂‘(𝑀 · 𝑋))‘𝑊) = 𝑊)
126		fniniseg 7074	. . . . . . 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 711	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}))
129	128	snssd 4817	. . . 4 ⊢ (𝜑 → {𝑊} ⊆ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}))
130		hashsng 14370	. . . . . . 7 ⊢ (𝑊 ∈ 𝐾 → (♯‘{𝑊}) = 1)
131	78, 130	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘{𝑊}) = 1)
132		ssdomg 9029	. . . . . . . . . 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 9077	. . . . . . . . . 10 ⊢ {𝑊} ∈ Fin
135		hashdom 14380	. . . . . . . . . 10 ⊢ (({𝑊} ∈ Fin ∧ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}) ∈ V) → ((♯‘{𝑊}) ≤ (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ↔ {𝑊} ≼ (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})))
136	134, 33, 135	mp2an 690	. . . . . . . . 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 5176	. . . . . . 7 ⊢ (𝜑 → 1 ≤ (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})))
139		hashcl 14357	. . . . . . . . . 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 12573	. . . . . . . 8 ⊢ (𝜑 → (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ∈ ℝ)
142		1re 11254	. . . . . . . 8 ⊢ 1 ∈ ℝ
143		letri3 11339	. . . . . . . 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 711	. . . . . 6 ⊢ (𝜑 → (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) = 1)
146	131, 145	eqtr4d 2771	. . . . 5 ⊢ (𝜑 → (♯‘{𝑊}) = (♯‘(^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})))
147		hashen 14348	. . . . . 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 9290	. . . 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 1368	. . 3 ⊢ (𝜑 → {𝑊} = (^◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊}))
152	30, 151	eleqtrrd 2832	. 2 ⊢ (𝜑 → 𝑁 ∈ {𝑊})
153		elsni 4649	. 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 1533 ∈ wcel 2098 ≠ wne 2937 Vcvv 3473 ∖ cdif 3946 ⊆ wss 3949 {csn 4632 class class class wbr 5152 I cid 5579 × cxp 5680 ^◡ccnv 5681 ↾ cres 5684 “ cima 5685 Fn wfn 6548 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 ∘_f cof 7690 ≈ cen 8969 ≼ cdom 8970 Fincfn 8972 ℝcr 11147 1c1 11149 ≤ cle 11289 ℕ₀cn0 12512 ♯chash 14331 Basecbs 17189 ._rcmulr 17243 Scalarcsca 17245 ·_𝑠 cvsca 17246 0_gc0g 17430 ↑_s cpws 17437 ._gcmg 19037 mulGrpcmgp 20088 Ringcrg 20187 CRingccrg 20188 RingHom crh 20422 AssAlgcasa 21798 algSccascl 21800 var₁cv1 22113 Poly₁cpl1 22114 coe₁cco1 22115 eval₁ce1 22252 deg₁ cdg1 26015

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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 ax-addf 11227

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-ofr 7693 df-om 7879 df-1st 8001 df-2nd 8002 df-supp 8174 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-oadd 8499 df-er 8733 df-map 8855 df-pm 8856 df-ixp 8925 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-fsupp 9396 df-sup 9475 df-oi 9543 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-xnn0 12585 df-z 12599 df-dec 12718 df-uz 12863 df-fz 13527 df-fzo 13670 df-seq 14009 df-hash 14332 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-starv 17257 df-sca 17258 df-vsca 17259 df-ip 17260 df-tset 17261 df-ple 17262 df-ds 17264 df-unif 17265 df-hom 17266 df-cco 17267 df-0g 17432 df-gsum 17433 df-prds 17438 df-pws 17440 df-mre 17575 df-mrc 17576 df-acs 17578 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-mhm 18749 df-submnd 18750 df-grp 18907 df-minusg 18908 df-sbg 18909 df-mulg 19038 df-subg 19092 df-ghm 19182 df-cntz 19282 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-srg 20141 df-ring 20189 df-cring 20190 df-rhm 20425 df-subrng 20497 df-subrg 20522 df-lmod 20759 df-lss 20830 df-lsp 20870 df-cnfld 21294 df-assa 21801 df-asp 21802 df-ascl 21803 df-psr 21856 df-mvr 21857 df-mpl 21858 df-opsr 21860 df-evls 22035 df-evl 22036 df-psr1 22117 df-vr1 22118 df-ply1 22119 df-coe1 22120 df-evl1 22254 df-mdeg 26016 df-deg1 26017

This theorem is referenced by: fta1b 26134

W3C validator