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Theorem dchrptlem2 27310

Description: Lemma for dchrpt 27312. (Contributed by Mario Carneiro, 28-Apr-2016.)

**Hypotheses**
Ref	Expression
dchrpt.g	⊢ 𝐺 = (DChr‘𝑁)
dchrpt.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
dchrpt.d	⊢ 𝐷 = (Base‘𝐺)
dchrpt.b	⊢ 𝐵 = (Base‘𝑍)
dchrpt.1	⊢ 1 = (1_r‘𝑍)
dchrpt.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
dchrpt.n1	⊢ (𝜑 → 𝐴 ≠ 1 )
dchrpt.u	⊢ 𝑈 = (Unit‘𝑍)
dchrpt.h	⊢ 𝐻 = ((mulGrp‘𝑍) ↾_s 𝑈)
dchrpt.m	⊢ · = (._g‘𝐻)
dchrpt.s	⊢ 𝑆 = (𝑘 ∈ dom 𝑊 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))))
dchrpt.au	⊢ (𝜑 → 𝐴 ∈ 𝑈)
dchrpt.w	⊢ (𝜑 → 𝑊 ∈ Word 𝑈)
dchrpt.2	⊢ (𝜑 → 𝐻dom DProd 𝑆)
dchrpt.3	⊢ (𝜑 → (𝐻 DProd 𝑆) = 𝑈)
dchrpt.p	⊢ 𝑃 = (𝐻dProj𝑆)
dchrpt.o	⊢ 𝑂 = (od‘𝐻)
dchrpt.t	⊢ 𝑇 = (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))
dchrpt.i	⊢ (𝜑 → 𝐼 ∈ dom 𝑊)
dchrpt.4	⊢ (𝜑 → ((𝑃‘𝐼)‘𝐴) ≠ 1 )
dchrpt.5	⊢ 𝑋 = (𝑢 ∈ 𝑈 ↦ (℩ℎ∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝑢) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚))))

**Assertion**
Ref	Expression
dchrptlem2	⊢ (𝜑 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1)

Distinct variable groups: ℎ,𝑘,𝑚,𝑛,𝑥, 1 𝑢,ℎ,𝐴,𝑘,𝑚,𝑛,𝑥 ℎ,𝐼,𝑘,𝑚,𝑢 𝑥,𝐵 𝑥,𝐺 ℎ,𝐻,𝑘,𝑚,𝑛,𝑢,𝑥 𝑥,𝑁 ℎ,𝑊,𝑘,𝑚,𝑛,𝑢,𝑥 · ,ℎ,𝑘,𝑚,𝑛,𝑢,𝑥 𝑥,𝑋 𝑃,ℎ,𝑚,𝑢 𝑆,ℎ,𝑘,𝑚,𝑛,𝑢,𝑥 ℎ,𝑍,𝑘,𝑚,𝑛,𝑢,𝑥 𝑥,𝐷 𝜑,ℎ,𝑘,𝑚,𝑛,𝑥 𝑇,ℎ,𝑚,𝑢 𝑈,ℎ,𝑚,𝑢,𝑥 Allowed substitution hints: 𝜑(𝑢) 𝐵(𝑢,ℎ,𝑘,𝑚,𝑛) 𝐷(𝑢,ℎ,𝑘,𝑚,𝑛) 𝑃(𝑥,𝑘,𝑛) 𝑇(𝑥,𝑘,𝑛) 𝑈(𝑘,𝑛) 1 (𝑢) 𝐺(𝑢,ℎ,𝑘,𝑚,𝑛) 𝐼(𝑥,𝑛) 𝑁(𝑢,ℎ,𝑘,𝑚,𝑛) 𝑂(𝑥,𝑢,ℎ,𝑘,𝑚,𝑛) 𝑋(𝑢,ℎ,𝑘,𝑚,𝑛)

Proof of Theorem dchrptlem2 Dummy variables 𝑎 𝑏 𝑣 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dchrpt.g	. . 3 ⊢ 𝐺 = (DChr‘𝑁)
2		dchrpt.z	. . 3 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
3		dchrpt.b	. . 3 ⊢ 𝐵 = (Base‘𝑍)
4		dchrpt.u	. . 3 ⊢ 𝑈 = (Unit‘𝑍)
5		dchrpt.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ)
6		dchrpt.d	. . 3 ⊢ 𝐷 = (Base‘𝐺)
7		fveq2 6905	. . 3 ⊢ (𝑣 = 𝑥 → (𝑋‘𝑣) = (𝑋‘𝑥))
8		fveq2 6905	. . 3 ⊢ (𝑣 = 𝑦 → (𝑋‘𝑣) = (𝑋‘𝑦))
9		fveq2 6905	. . 3 ⊢ (𝑣 = (𝑥(._r‘𝑍)𝑦) → (𝑋‘𝑣) = (𝑋‘(𝑥(._r‘𝑍)𝑦)))
10		fveq2 6905	. . 3 ⊢ (𝑣 = (1_r‘𝑍) → (𝑋‘𝑣) = (𝑋‘(1_r‘𝑍)))
11		dchrpt.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐻dom DProd 𝑆)
12		zex 12624	. . . . . . . . . . . . 13 ⊢ ℤ ∈ V
13	12	mptex 7244	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) ∈ V
14	13	rnex 7933	. . . . . . . . . . 11 ⊢ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) ∈ V
15		dchrpt.s	. . . . . . . . . . 11 ⊢ 𝑆 = (𝑘 ∈ dom 𝑊 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))))
16	14, 15	dmmpti 6711	. . . . . . . . . 10 ⊢ dom 𝑆 = dom 𝑊
17	16	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑆 = dom 𝑊)
18		dchrpt.p	. . . . . . . . 9 ⊢ 𝑃 = (𝐻dProj𝑆)
19		dchrpt.i	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ dom 𝑊)
20	11, 17, 18, 19	dpjf 20078	. . . . . . . 8 ⊢ (𝜑 → (𝑃‘𝐼):(𝐻 DProd 𝑆)⟶(𝑆‘𝐼))
21		dchrpt.3	. . . . . . . . 9 ⊢ (𝜑 → (𝐻 DProd 𝑆) = 𝑈)
22	21	feq2d 6721	. . . . . . . 8 ⊢ (𝜑 → ((𝑃‘𝐼):(𝐻 DProd 𝑆)⟶(𝑆‘𝐼) ↔ (𝑃‘𝐼):𝑈⟶(𝑆‘𝐼)))
23	20, 22	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (𝑃‘𝐼):𝑈⟶(𝑆‘𝐼))
24	23	ffvelcdmda 7103	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑈) → ((𝑃‘𝐼)‘𝑣) ∈ (𝑆‘𝐼))
25	19	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑈) → 𝐼 ∈ dom 𝑊)
26		oveq1 7439	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑎 → (𝑛 · (𝑊‘𝑘)) = (𝑎 · (𝑊‘𝑘)))
27	26	cbvmptv 5254	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) = (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝑘)))
28		fveq2 6905	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐼 → (𝑊‘𝑘) = (𝑊‘𝐼))
29	28	oveq2d 7448	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐼 → (𝑎 · (𝑊‘𝑘)) = (𝑎 · (𝑊‘𝐼)))
30	29	mpteq2dv 5243	. . . . . . . . . 10 ⊢ (𝑘 = 𝐼 → (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝑘))) = (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))))
31	27, 30	eqtrid 2788	. . . . . . . . 9 ⊢ (𝑘 = 𝐼 → (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) = (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))))
32	31	rneqd 5948	. . . . . . . 8 ⊢ (𝑘 = 𝐼 → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) = ran (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))))
33	32, 15, 14	fvmpt3i 7020	. . . . . . 7 ⊢ (𝐼 ∈ dom 𝑊 → (𝑆‘𝐼) = ran (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))))
34	25, 33	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑈) → (𝑆‘𝐼) = ran (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))))
35	24, 34	eleqtrd 2842	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑈) → ((𝑃‘𝐼)‘𝑣) ∈ ran (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))))
36		eqid 2736	. . . . . 6 ⊢ (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))) = (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼)))
37		ovex 7465	. . . . . 6 ⊢ (𝑎 · (𝑊‘𝐼)) ∈ V
38	36, 37	elrnmpti 5972	. . . . 5 ⊢ (((𝑃‘𝐼)‘𝑣) ∈ ran (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))) ↔ ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))
39	35, 38	sylib 218	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑈) → ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))
40		dchrpt.1	. . . . . 6 ⊢ 1 = (1_r‘𝑍)
41		dchrpt.n1	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 1 )
42		dchrpt.h	. . . . . 6 ⊢ 𝐻 = ((mulGrp‘𝑍) ↾_s 𝑈)
43		dchrpt.m	. . . . . 6 ⊢ · = (._g‘𝐻)
44		dchrpt.au	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
45		dchrpt.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Word 𝑈)
46		dchrpt.o	. . . . . 6 ⊢ 𝑂 = (od‘𝐻)
47		dchrpt.t	. . . . . 6 ⊢ 𝑇 = (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))
48		dchrpt.4	. . . . . 6 ⊢ (𝜑 → ((𝑃‘𝐼)‘𝐴) ≠ 1 )
49		dchrpt.5	. . . . . 6 ⊢ 𝑋 = (𝑢 ∈ 𝑈 ↦ (℩ℎ∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝑢) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚))))
50	1, 2, 6, 3, 40, 5, 41, 4, 42, 43, 15, 44, 45, 11, 21, 18, 46, 47, 19, 48, 49	dchrptlem1 27309	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))) → (𝑋‘𝑣) = (𝑇↑𝑎))
51		neg1cn 12381	. . . . . . . . 9 ⊢ -1 ∈ ℂ
52		2re 12341	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
53	5	nnnn0d 12589	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
54	2	zncrng 21564	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
55		crngring 20243	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
56	53, 54, 55	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ Ring)
57	4, 42	unitgrp 20384	. . . . . . . . . . . . 13 ⊢ (𝑍 ∈ Ring → 𝐻 ∈ Grp)
58	56, 57	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ∈ Grp)
59	2, 3	znfi 21579	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 𝐵 ∈ Fin)
60	5, 59	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ Fin)
61	3, 4	unitss 20377	. . . . . . . . . . . . 13 ⊢ 𝑈 ⊆ 𝐵
62		ssfi 9214	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ Fin ∧ 𝑈 ⊆ 𝐵) → 𝑈 ∈ Fin)
63	60, 61, 62	sylancl 586	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ Fin)
64		wrdf 14558	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ Word 𝑈 → 𝑊:(0..^(♯‘𝑊))⟶𝑈)
65	45, 64	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑊:(0..^(♯‘𝑊))⟶𝑈)
66	65	fdmd 6745	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
67	19, 66	eleqtrd 2842	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ (0..^(♯‘𝑊)))
68	65, 67	ffvelcdmd 7104	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑊‘𝐼) ∈ 𝑈)
69	4, 42	unitgrpbas 20383	. . . . . . . . . . . . 13 ⊢ 𝑈 = (Base‘𝐻)
70	69, 46	odcl2 19584	. . . . . . . . . . . 12 ⊢ ((𝐻 ∈ Grp ∧ 𝑈 ∈ Fin ∧ (𝑊‘𝐼) ∈ 𝑈) → (𝑂‘(𝑊‘𝐼)) ∈ ℕ)
71	58, 63, 68, 70	syl3anc 1372	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂‘(𝑊‘𝐼)) ∈ ℕ)
72		nndivre 12308	. . . . . . . . . . 11 ⊢ ((2 ∈ ℝ ∧ (𝑂‘(𝑊‘𝐼)) ∈ ℕ) → (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℝ)
73	52, 71, 72	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℝ)
74	73	recnd 11290	. . . . . . . . 9 ⊢ (𝜑 → (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℂ)
75		cxpcl 26717	. . . . . . . . 9 ⊢ ((-1 ∈ ℂ ∧ (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℂ) → (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ∈ ℂ)
76	51, 74, 75	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ∈ ℂ)
77	47, 76	eqeltrid 2844	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ ℂ)
78	77	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))) → 𝑇 ∈ ℂ)
79	51	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → -1 ∈ ℂ)
80		neg1ne0 12383	. . . . . . . . . 10 ⊢ -1 ≠ 0
81	80	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → -1 ≠ 0)
82	79, 81, 74	cxpne0d 26756	. . . . . . . 8 ⊢ (𝜑 → (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ≠ 0)
83	47	neeq1i 3004	. . . . . . . 8 ⊢ (𝑇 ≠ 0 ↔ (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ≠ 0)
84	82, 83	sylibr 234	. . . . . . 7 ⊢ (𝜑 → 𝑇 ≠ 0)
85	84	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))) → 𝑇 ≠ 0)
86		simprl 770	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))) → 𝑎 ∈ ℤ)
87	78, 85, 86	expclzd 14192	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))) → (𝑇↑𝑎) ∈ ℂ)
88	50, 87	eqeltrd 2840	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))) → (𝑋‘𝑣) ∈ ℂ)
89	39, 88	rexlimddv 3160	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑈) → (𝑋‘𝑣) ∈ ℂ)
90		fveqeq2 6914	. . . . . 6 ⊢ (𝑣 = 𝑥 → (((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼))))
91	90	rexbidv 3178	. . . . 5 ⊢ (𝑣 = 𝑥 → (∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼))))
92	39	ralrimiva 3145	. . . . . 6 ⊢ (𝜑 → ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))
93	92	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))
94		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝑈)
95	91, 93, 94	rspcdva 3622	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)))
96		fveqeq2 6914	. . . . . . 7 ⊢ (𝑣 = 𝑦 → (((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ((𝑃‘𝐼)‘𝑦) = (𝑎 · (𝑊‘𝐼))))
97	96	rexbidv 3178	. . . . . 6 ⊢ (𝑣 = 𝑦 → (∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑦) = (𝑎 · (𝑊‘𝐼))))
98		oveq1 7439	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (𝑎 · (𝑊‘𝐼)) = (𝑏 · (𝑊‘𝐼)))
99	98	eqeq2d 2747	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (((𝑃‘𝐼)‘𝑦) = (𝑎 · (𝑊‘𝐼)) ↔ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))
100	99	cbvrexvw 3237	. . . . . 6 ⊢ (∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑦) = (𝑎 · (𝑊‘𝐼)) ↔ ∃𝑏 ∈ ℤ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼)))
101	97, 100	bitrdi 287	. . . . 5 ⊢ (𝑣 = 𝑦 → (∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ∃𝑏 ∈ ℤ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))
102		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝑈)
103	101, 93, 102	rspcdva 3622	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ∃𝑏 ∈ ℤ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼)))
104		reeanv 3228	. . . . 5 ⊢ (∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))) ↔ (∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ∃𝑏 ∈ ℤ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))
105	77	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))) → 𝑇 ∈ ℂ)
106	84	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))) → 𝑇 ≠ 0)
107		simprll 778	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))) → 𝑎 ∈ ℤ)
108		simprlr 779	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))) → 𝑏 ∈ ℤ)
109		expaddz 14148	. . . . . . . . 9 ⊢ (((𝑇 ∈ ℂ ∧ 𝑇 ≠ 0) ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → (𝑇↑(𝑎 + 𝑏)) = ((𝑇↑𝑎) · (𝑇↑𝑏)))
110	105, 106, 107, 108, 109	syl22anc 838	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))) → (𝑇↑(𝑎 + 𝑏)) = ((𝑇↑𝑎) · (𝑇↑𝑏)))
111		simpll 766	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))) → 𝜑)
112	56	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))) → 𝑍 ∈ Ring)
113	94	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))) → 𝑥 ∈ 𝑈)
114	102	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))) → 𝑦 ∈ 𝑈)
115		eqid 2736	. . . . . . . . . . 11 ⊢ (._r‘𝑍) = (._r‘𝑍)
116	4, 115	unitmulcl 20381	. . . . . . . . . 10 ⊢ ((𝑍 ∈ Ring ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈) → (𝑥(._r‘𝑍)𝑦) ∈ 𝑈)
117	112, 113, 114, 116	syl3anc 1372	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))) → (𝑥(._r‘𝑍)𝑦) ∈ 𝑈)
118	107, 108	zaddcld 12728	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))) → (𝑎 + 𝑏) ∈ ℤ)
119		simprrl 780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))) → ((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)))
120		simprrr 781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))) → ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼)))
121	119, 120	oveq12d 7450	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))) → (((𝑃‘𝐼)‘𝑥)(._r‘𝑍)((𝑃‘𝐼)‘𝑦)) = ((𝑎 · (𝑊‘𝐼))(._r‘𝑍)(𝑏 · (𝑊‘𝐼))))
122	11, 17, 18, 19	dpjghm 20084	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃‘𝐼) ∈ ((𝐻 ↾_s (𝐻 DProd 𝑆)) GrpHom 𝐻))
123	21	oveq2d 7448	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐻 ↾_s (𝐻 DProd 𝑆)) = (𝐻 ↾_s 𝑈))
124	42	ovexi 7466	. . . . . . . . . . . . . . . 16 ⊢ 𝐻 ∈ V
125	69	ressid 17291	. . . . . . . . . . . . . . . 16 ⊢ (𝐻 ∈ V → (𝐻 ↾_s 𝑈) = 𝐻)
126	124, 125	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (𝐻 ↾_s 𝑈) = 𝐻
127	123, 126	eqtrdi 2792	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐻 ↾_s (𝐻 DProd 𝑆)) = 𝐻)
128	127	oveq1d 7447	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐻 ↾_s (𝐻 DProd 𝑆)) GrpHom 𝐻) = (𝐻 GrpHom 𝐻))
129	122, 128	eleqtrd 2842	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃‘𝐼) ∈ (𝐻 GrpHom 𝐻))
130	129	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))) → (𝑃‘𝐼) ∈ (𝐻 GrpHom 𝐻))
131	4	fvexi 6919	. . . . . . . . . . . . 13 ⊢ 𝑈 ∈ V
132		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
133	132, 115	mgpplusg 20142	. . . . . . . . . . . . . 14 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
134	42, 133	ressplusg 17335	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ V → (._r‘𝑍) = (+_g‘𝐻))
135	131, 134	ax-mp 5	. . . . . . . . . . . 12 ⊢ (._r‘𝑍) = (+_g‘𝐻)
136	69, 135, 135	ghmlin 19240	. . . . . . . . . . 11 ⊢ (((𝑃‘𝐼) ∈ (𝐻 GrpHom 𝐻) ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈) → ((𝑃‘𝐼)‘(𝑥(._r‘𝑍)𝑦)) = (((𝑃‘𝐼)‘𝑥)(._r‘𝑍)((𝑃‘𝐼)‘𝑦)))
137	130, 113, 114, 136	syl3anc 1372	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))) → ((𝑃‘𝐼)‘(𝑥(._r‘𝑍)𝑦)) = (((𝑃‘𝐼)‘𝑥)(._r‘𝑍)((𝑃‘𝐼)‘𝑦)))
138	58	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))) → 𝐻 ∈ Grp)
139	68	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))) → (𝑊‘𝐼) ∈ 𝑈)
140	69, 43, 135	mulgdir 19125	. . . . . . . . . . 11 ⊢ ((𝐻 ∈ Grp ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ (𝑊‘𝐼) ∈ 𝑈)) → ((𝑎 + 𝑏) · (𝑊‘𝐼)) = ((𝑎 · (𝑊‘𝐼))(._r‘𝑍)(𝑏 · (𝑊‘𝐼))))
141	138, 107, 108, 139, 140	syl13anc 1373	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))) → ((𝑎 + 𝑏) · (𝑊‘𝐼)) = ((𝑎 · (𝑊‘𝐼))(._r‘𝑍)(𝑏 · (𝑊‘𝐼))))
142	121, 137, 141	3eqtr4d 2786	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))) → ((𝑃‘𝐼)‘(𝑥(._r‘𝑍)𝑦)) = ((𝑎 + 𝑏) · (𝑊‘𝐼)))
143	1, 2, 6, 3, 40, 5, 41, 4, 42, 43, 15, 44, 45, 11, 21, 18, 46, 47, 19, 48, 49	dchrptlem1 27309	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥(._r‘𝑍)𝑦) ∈ 𝑈) ∧ ((𝑎 + 𝑏) ∈ ℤ ∧ ((𝑃‘𝐼)‘(𝑥(._r‘𝑍)𝑦)) = ((𝑎 + 𝑏) · (𝑊‘𝐼)))) → (𝑋‘(𝑥(._r‘𝑍)𝑦)) = (𝑇↑(𝑎 + 𝑏)))
144	111, 117, 118, 142, 143	syl22anc 838	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))) → (𝑋‘(𝑥(._r‘𝑍)𝑦)) = (𝑇↑(𝑎 + 𝑏)))
145	1, 2, 6, 3, 40, 5, 41, 4, 42, 43, 15, 44, 45, 11, 21, 18, 46, 47, 19, 48, 49	dchrptlem1 27309	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)))) → (𝑋‘𝑥) = (𝑇↑𝑎))
146	111, 113, 107, 119, 145	syl22anc 838	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))) → (𝑋‘𝑥) = (𝑇↑𝑎))
147	1, 2, 6, 3, 40, 5, 41, 4, 42, 43, 15, 44, 45, 11, 21, 18, 46, 47, 19, 48, 49	dchrptlem1 27309	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑈) ∧ (𝑏 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼)))) → (𝑋‘𝑦) = (𝑇↑𝑏))
148	111, 114, 108, 120, 147	syl22anc 838	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))) → (𝑋‘𝑦) = (𝑇↑𝑏))
149	146, 148	oveq12d 7450	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))) → ((𝑋‘𝑥) · (𝑋‘𝑦)) = ((𝑇↑𝑎) · (𝑇↑𝑏)))
150	110, 144, 149	3eqtr4d 2786	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))) → (𝑋‘(𝑥(._r‘𝑍)𝑦)) = ((𝑋‘𝑥) · (𝑋‘𝑦)))
151	150	expr 456	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → ((((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))) → (𝑋‘(𝑥(._r‘𝑍)𝑦)) = ((𝑋‘𝑥) · (𝑋‘𝑦))))
152	151	rexlimdvva 3212	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ (((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))) → (𝑋‘(𝑥(._r‘𝑍)𝑦)) = ((𝑋‘𝑥) · (𝑋‘𝑦))))
153	104, 152	biimtrrid 243	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ((∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)) ∧ ∃𝑏 ∈ ℤ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))) → (𝑋‘(𝑥(._r‘𝑍)𝑦)) = ((𝑋‘𝑥) · (𝑋‘𝑦))))
154	95, 103, 153	mp2and 699	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑋‘(𝑥(._r‘𝑍)𝑦)) = ((𝑋‘𝑥) · (𝑋‘𝑦)))
155		id 22	. . . . 5 ⊢ (𝜑 → 𝜑)
156		eqid 2736	. . . . . . 7 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
157	4, 156	1unit 20375	. . . . . 6 ⊢ (𝑍 ∈ Ring → (1_r‘𝑍) ∈ 𝑈)
158	56, 157	syl 17	. . . . 5 ⊢ (𝜑 → (1_r‘𝑍) ∈ 𝑈)
159		0zd 12627	. . . . 5 ⊢ (𝜑 → 0 ∈ ℤ)
160		eqid 2736	. . . . . . . 8 ⊢ (0_g‘𝐻) = (0_g‘𝐻)
161	160, 160	ghmid 19241	. . . . . . 7 ⊢ ((𝑃‘𝐼) ∈ (𝐻 GrpHom 𝐻) → ((𝑃‘𝐼)‘(0_g‘𝐻)) = (0_g‘𝐻))
162	129, 161	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝑃‘𝐼)‘(0_g‘𝐻)) = (0_g‘𝐻))
163	4, 42, 156	unitgrpid 20386	. . . . . . . 8 ⊢ (𝑍 ∈ Ring → (1_r‘𝑍) = (0_g‘𝐻))
164	56, 163	syl 17	. . . . . . 7 ⊢ (𝜑 → (1_r‘𝑍) = (0_g‘𝐻))
165	164	fveq2d 6909	. . . . . 6 ⊢ (𝜑 → ((𝑃‘𝐼)‘(1_r‘𝑍)) = ((𝑃‘𝐼)‘(0_g‘𝐻)))
166	69, 160, 43	mulg0 19093	. . . . . . 7 ⊢ ((𝑊‘𝐼) ∈ 𝑈 → (0 · (𝑊‘𝐼)) = (0_g‘𝐻))
167	68, 166	syl 17	. . . . . 6 ⊢ (𝜑 → (0 · (𝑊‘𝐼)) = (0_g‘𝐻))
168	162, 165, 167	3eqtr4d 2786	. . . . 5 ⊢ (𝜑 → ((𝑃‘𝐼)‘(1_r‘𝑍)) = (0 · (𝑊‘𝐼)))
169	1, 2, 6, 3, 40, 5, 41, 4, 42, 43, 15, 44, 45, 11, 21, 18, 46, 47, 19, 48, 49	dchrptlem1 27309	. . . . 5 ⊢ (((𝜑 ∧ (1_r‘𝑍) ∈ 𝑈) ∧ (0 ∈ ℤ ∧ ((𝑃‘𝐼)‘(1_r‘𝑍)) = (0 · (𝑊‘𝐼)))) → (𝑋‘(1_r‘𝑍)) = (𝑇↑0))
170	155, 158, 159, 168, 169	syl22anc 838	. . . 4 ⊢ (𝜑 → (𝑋‘(1_r‘𝑍)) = (𝑇↑0))
171	77	exp0d 14181	. . . 4 ⊢ (𝜑 → (𝑇↑0) = 1)
172	170, 171	eqtrd 2776	. . 3 ⊢ (𝜑 → (𝑋‘(1_r‘𝑍)) = 1)
173	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 89, 154, 172	dchrelbasd 27284	. 2 ⊢ (𝜑 → (𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0)) ∈ 𝐷)
174	61, 44	sselid 3980	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
175		eleq1 2828	. . . . . . 7 ⊢ (𝑣 = 𝐴 → (𝑣 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈))
176		fveq2 6905	. . . . . . 7 ⊢ (𝑣 = 𝐴 → (𝑋‘𝑣) = (𝑋‘𝐴))
177	175, 176	ifbieq1d 4549	. . . . . 6 ⊢ (𝑣 = 𝐴 → if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0) = if(𝐴 ∈ 𝑈, (𝑋‘𝐴), 0))
178		eqid 2736	. . . . . 6 ⊢ (𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0)) = (𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0))
179		fvex 6918	. . . . . . 7 ⊢ (𝑋‘𝑣) ∈ V
180		c0ex 11256	. . . . . . 7 ⊢ 0 ∈ V
181	179, 180	ifex 4575	. . . . . 6 ⊢ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0) ∈ V
182	177, 178, 181	fvmpt3i 7020	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → ((𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0))‘𝐴) = if(𝐴 ∈ 𝑈, (𝑋‘𝐴), 0))
183	174, 182	syl 17	. . . 4 ⊢ (𝜑 → ((𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0))‘𝐴) = if(𝐴 ∈ 𝑈, (𝑋‘𝐴), 0))
184	44	iftrued 4532	. . . 4 ⊢ (𝜑 → if(𝐴 ∈ 𝑈, (𝑋‘𝐴), 0) = (𝑋‘𝐴))
185	183, 184	eqtrd 2776	. . 3 ⊢ (𝜑 → ((𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0))‘𝐴) = (𝑋‘𝐴))
186		fveqeq2 6914	. . . . . 6 ⊢ (𝑣 = 𝐴 → (((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼))))
187	186	rexbidv 3178	. . . . 5 ⊢ (𝑣 = 𝐴 → (∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼))))
188	187, 92, 44	rspcdva 3622	. . . 4 ⊢ (𝜑 → ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))
189	1, 2, 6, 3, 40, 5, 41, 4, 42, 43, 15, 44, 45, 11, 21, 18, 46, 47, 19, 48, 49	dchrptlem1 27309	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → (𝑋‘𝐴) = (𝑇↑𝑎))
190	47	oveq1i 7442	. . . . . . . 8 ⊢ (𝑇↑𝑎) = ((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑𝑎)
191	189, 190	eqtrdi 2792	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → (𝑋‘𝐴) = ((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑𝑎))
192	48	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → ((𝑃‘𝐼)‘𝐴) ≠ 1 )
193	58	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → 𝐻 ∈ Grp)
194	68	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → (𝑊‘𝐼) ∈ 𝑈)
195		simprl 770	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → 𝑎 ∈ ℤ)
196	69, 46, 43, 160	oddvds 19566	. . . . . . . . . . 11 ⊢ ((𝐻 ∈ Grp ∧ (𝑊‘𝐼) ∈ 𝑈 ∧ 𝑎 ∈ ℤ) → ((𝑂‘(𝑊‘𝐼)) ∥ 𝑎 ↔ (𝑎 · (𝑊‘𝐼)) = (0_g‘𝐻)))
197	193, 194, 195, 196	syl3anc 1372	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → ((𝑂‘(𝑊‘𝐼)) ∥ 𝑎 ↔ (𝑎 · (𝑊‘𝐼)) = (0_g‘𝐻)))
198	71	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → (𝑂‘(𝑊‘𝐼)) ∈ ℕ)
199		root1eq1 26799	. . . . . . . . . . 11 ⊢ (((𝑂‘(𝑊‘𝐼)) ∈ ℕ ∧ 𝑎 ∈ ℤ) → (((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑𝑎) = 1 ↔ (𝑂‘(𝑊‘𝐼)) ∥ 𝑎))
200	198, 195, 199	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → (((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑𝑎) = 1 ↔ (𝑂‘(𝑊‘𝐼)) ∥ 𝑎))
201		simprr 772	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))
202	40, 164	eqtrid 2788	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 = (0_g‘𝐻))
203	202	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → 1 = (0_g‘𝐻))
204	201, 203	eqeq12d 2752	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → (((𝑃‘𝐼)‘𝐴) = 1 ↔ (𝑎 · (𝑊‘𝐼)) = (0_g‘𝐻)))
205	197, 200, 204	3bitr4d 311	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → (((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑𝑎) = 1 ↔ ((𝑃‘𝐼)‘𝐴) = 1 ))
206	205	necon3bid 2984	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → (((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑𝑎) ≠ 1 ↔ ((𝑃‘𝐼)‘𝐴) ≠ 1 ))
207	192, 206	mpbird 257	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → ((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑𝑎) ≠ 1)
208	191, 207	eqnetrd 3007	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → (𝑋‘𝐴) ≠ 1)
209	208	rexlimdvaa 3155	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)) → (𝑋‘𝐴) ≠ 1))
210	44, 209	mpdan 687	. . . 4 ⊢ (𝜑 → (∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)) → (𝑋‘𝐴) ≠ 1))
211	188, 210	mpd 15	. . 3 ⊢ (𝜑 → (𝑋‘𝐴) ≠ 1)
212	185, 211	eqnetrd 3007	. 2 ⊢ (𝜑 → ((𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0))‘𝐴) ≠ 1)
213		fveq1 6904	. . . 4 ⊢ (𝑥 = (𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0)) → (𝑥‘𝐴) = ((𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0))‘𝐴))
214	213	neeq1d 2999	. . 3 ⊢ (𝑥 = (𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0)) → ((𝑥‘𝐴) ≠ 1 ↔ ((𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0))‘𝐴) ≠ 1))
215	214	rspcev 3621	. 2 ⊢ (((𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0)) ∈ 𝐷 ∧ ((𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0))‘𝐴) ≠ 1) → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1)
216	173, 212, 215	syl2anc 584	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 Vcvv 3479 ⊆ wss 3950 ifcif 4524 class class class wbr 5142 ↦ cmpt 5224 dom cdm 5684 ran crn 5685 ℩cio 6511 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 Fincfn 8986 ℂcc 11154 ℝcr 11155 0cc0 11156 1c1 11157 + caddc 11159 · cmul 11161 -cneg 11494 / cdiv 11921 ℕcn 12267 2c2 12322 ℕ₀cn0 12528 ℤcz 12615 ..^cfzo 13695 ↑cexp 14103 ♯chash 14370 Word cword 14553 ∥ cdvds 16291 Basecbs 17248 ↾_s cress 17275 +_gcplusg 17298 ._rcmulr 17299 0_gc0g 17485 Grpcgrp 18952 ._gcmg 19086 GrpHom cghm 19231 odcod 19543 DProd cdprd 20014 dProjcdpj 20015 mulGrpcmgp 20138 1_rcur 20179 Ringcrg 20231 CRingccrg 20232 Unitcui 20356 ℤ/nℤczn 21514 ↑_𝑐ccxp 26598 DChrcdchr 27277

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235 ax-mulf 11236

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-tpos 8252 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-omul 8512 df-er 8746 df-ec 8748 df-qs 8752 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-acn 9983 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-seq 14044 df-exp 14104 df-fac 14314 df-bc 14343 df-hash 14371 df-word 14554 df-shft 15107 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-limsup 15508 df-clim 15525 df-rlim 15526 df-sum 15724 df-ef 16104 df-sin 16106 df-cos 16107 df-pi 16109 df-dvds 16292 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-qus 17555 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-mulg 19087 df-subg 19142 df-nsg 19143 df-eqg 19144 df-ghm 19232 df-gim 19278 df-cntz 19336 df-oppg 19365 df-od 19547 df-lsm 19655 df-pj1 19656 df-cmn 19801 df-abl 19802 df-dprd 20016 df-dpj 20017 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-cring 20234 df-oppr 20335 df-dvdsr 20358 df-unit 20359 df-rhm 20473 df-subrng 20547 df-subrg 20571 df-lmod 20861 df-lss 20931 df-lsp 20971 df-sra 21173 df-rgmod 21174 df-lidl 21219 df-rsp 21220 df-2idl 21261 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-cnfld 21366 df-zring 21459 df-zrh 21515 df-zn 21518 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-lp 23145 df-perf 23146 df-cn 23236 df-cnp 23237 df-haus 23324 df-tx 23571 df-hmeo 23764 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-xms 24331 df-ms 24332 df-tms 24333 df-cncf 24905 df-limc 25902 df-dv 25903 df-log 26599 df-cxp 26600 df-dchr 27278

This theorem is referenced by: dchrptlem3 27311

W3C validator