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Theorem dchrptlem2 27246

Description: Lemma for dchrpt 27248. (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 6827	. . 3 ⊢ (𝑣 = 𝑥 → (𝑋‘𝑣) = (𝑋‘𝑥))
8		fveq2 6827	. . 3 ⊢ (𝑣 = 𝑦 → (𝑋‘𝑣) = (𝑋‘𝑦))
9		fveq2 6827	. . 3 ⊢ (𝑣 = (𝑥(._r‘𝑍)𝑦) → (𝑋‘𝑣) = (𝑋‘(𝑥(._r‘𝑍)𝑦)))
10		fveq2 6827	. . 3 ⊢ (𝑣 = (1_r‘𝑍) → (𝑋‘𝑣) = (𝑋‘(1_r‘𝑍)))
11		dchrpt.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐻dom DProd 𝑆)
12		zex 12524	. . . . . . . . . . . . 13 ⊢ ℤ ∈ V
13	12	mptex 7167	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) ∈ V
14	13	rnex 7850	. . . . . . . . . . 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 6629	. . . . . . . . . 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 20025	. . . . . . . 8 ⊢ (𝜑 → (𝑃‘𝐼):(𝐻 DProd 𝑆)⟶(𝑆‘𝐼))
21		dchrpt.3	. . . . . . . . 9 ⊢ (𝜑 → (𝐻 DProd 𝑆) = 𝑈)
22	21	feq2d 6639	. . . . . . . 8 ⊢ (𝜑 → ((𝑃‘𝐼):(𝐻 DProd 𝑆)⟶(𝑆‘𝐼) ↔ (𝑃‘𝐼):𝑈⟶(𝑆‘𝐼)))
23	20, 22	mpbid 233	. . . . . . 7 ⊢ (𝜑 → (𝑃‘𝐼):𝑈⟶(𝑆‘𝐼))
24	23	ffvelcdmda 7025	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑈) → ((𝑃‘𝐼)‘𝑣) ∈ (𝑆‘𝐼))
25	19	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑈) → 𝐼 ∈ dom 𝑊)
26		oveq1 7363	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑎 → (𝑛 · (𝑊‘𝑘)) = (𝑎 · (𝑊‘𝑘)))
27	26	cbvmptv 5176	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) = (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝑘)))
28		fveq2 6827	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐼 → (𝑊‘𝑘) = (𝑊‘𝐼))
29	28	oveq2d 7372	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐼 → (𝑎 · (𝑊‘𝑘)) = (𝑎 · (𝑊‘𝐼)))
30	29	mpteq2dv 5166	. . . . . . . . . 10 ⊢ (𝑘 = 𝐼 → (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝑘))) = (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))))
31	27, 30	eqtrid 2786	. . . . . . . . 9 ⊢ (𝑘 = 𝐼 → (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) = (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))))
32	31	rneqd 5880	. . . . . . . 8 ⊢ (𝑘 = 𝐼 → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) = ran (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))))
33	32, 15, 14	fvmpt3i 6941	. . . . . . 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 2841	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑈) → ((𝑃‘𝐼)‘𝑣) ∈ ran (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))))
36		eqid 2739	. . . . . 6 ⊢ (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))) = (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼)))
37		ovex 7389	. . . . . 6 ⊢ (𝑎 · (𝑊‘𝐼)) ∈ V
38	36, 37	elrnmpti 5904	. . . . 5 ⊢ (((𝑃‘𝐼)‘𝑣) ∈ ran (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))) ↔ ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))
39	35, 38	sylib 219	. . . 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 27245	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))) → (𝑋‘𝑣) = (𝑇↑𝑎))
51		neg1cn 12135	. . . . . . . . 9 ⊢ -1 ∈ ℂ
52		2re 12246	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
53	5	nnnn0d 12489	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
54	2	zncrng 21519	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
55		crngring 20217	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
56	53, 54, 55	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ Ring)
57	4, 42	unitgrp 20354	. . . . . . . . . . . . 13 ⊢ (𝑍 ∈ Ring → 𝐻 ∈ Grp)
58	56, 57	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ∈ Grp)
59	2, 3	znfi 21534	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 𝐵 ∈ Fin)
60	5, 59	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ Fin)
61	3, 4	unitss 20347	. . . . . . . . . . . . 13 ⊢ 𝑈 ⊆ 𝐵
62		ssfi 9097	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ Fin ∧ 𝑈 ⊆ 𝐵) → 𝑈 ∈ Fin)
63	60, 61, 62	sylancl 592	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ Fin)
64		wrdf 14471	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ Word 𝑈 → 𝑊:(0..^(♯‘𝑊))⟶𝑈)
65	45, 64	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑊:(0..^(♯‘𝑊))⟶𝑈)
66	65	fdmd 6665	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
67	19, 66	eleqtrd 2841	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ (0..^(♯‘𝑊)))
68	65, 67	ffvelcdmd 7026	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑊‘𝐼) ∈ 𝑈)
69	4, 42	unitgrpbas 20353	. . . . . . . . . . . . 13 ⊢ 𝑈 = (Base‘𝐻)
70	69, 46	odcl2 19531	. . . . . . . . . . . 12 ⊢ ((𝐻 ∈ Grp ∧ 𝑈 ∈ Fin ∧ (𝑊‘𝐼) ∈ 𝑈) → (𝑂‘(𝑊‘𝐼)) ∈ ℕ)
71	58, 63, 68, 70	syl3anc 1379	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂‘(𝑊‘𝐼)) ∈ ℕ)
72		nndivre 12209	. . . . . . . . . . 11 ⊢ ((2 ∈ ℝ ∧ (𝑂‘(𝑊‘𝐼)) ∈ ℕ) → (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℝ)
73	52, 71, 72	sylancr 593	. . . . . . . . . 10 ⊢ (𝜑 → (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℝ)
74	73	recnd 11164	. . . . . . . . 9 ⊢ (𝜑 → (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℂ)
75		cxpcl 26656	. . . . . . . . 9 ⊢ ((-1 ∈ ℂ ∧ (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℂ) → (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ∈ ℂ)
76	51, 74, 75	sylancr 593	. . . . . . . 8 ⊢ (𝜑 → (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ∈ ℂ)
77	47, 76	eqeltrid 2843	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ ℂ)
78	77	ad2antrr 732	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))) → 𝑇 ∈ ℂ)
79	51	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → -1 ∈ ℂ)
80		neg1ne0 12137	. . . . . . . . . 10 ⊢ -1 ≠ 0
81	80	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → -1 ≠ 0)
82	79, 81, 74	cxpne0d 26695	. . . . . . . 8 ⊢ (𝜑 → (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ≠ 0)
83	47	neeq1i 2998	. . . . . . . 8 ⊢ (𝑇 ≠ 0 ↔ (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ≠ 0)
84	82, 83	sylibr 235	. . . . . . 7 ⊢ (𝜑 → 𝑇 ≠ 0)
85	84	ad2antrr 732	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))) → 𝑇 ≠ 0)
86		simprl 776	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))) → 𝑎 ∈ ℤ)
87	78, 85, 86	expclzd 14104	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))) → (𝑇↑𝑎) ∈ ℂ)
88	50, 87	eqeltrd 2839	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))) → (𝑋‘𝑣) ∈ ℂ)
89	39, 88	rexlimddv 3146	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑈) → (𝑋‘𝑣) ∈ ℂ)
90		fveqeq2 6836	. . . . . 6 ⊢ (𝑣 = 𝑥 → (((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼))))
91	90	rexbidv 3163	. . . . 5 ⊢ (𝑣 = 𝑥 → (∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼))))
92	39	ralrimiva 3131	. . . . . 6 ⊢ (𝜑 → ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))
93	92	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))
94		simprl 776	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝑈)
95	91, 93, 94	rspcdva 3561	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)))
96		fveqeq2 6836	. . . . . . 7 ⊢ (𝑣 = 𝑦 → (((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ((𝑃‘𝐼)‘𝑦) = (𝑎 · (𝑊‘𝐼))))
97	96	rexbidv 3163	. . . . . 6 ⊢ (𝑣 = 𝑦 → (∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑦) = (𝑎 · (𝑊‘𝐼))))
98		oveq1 7363	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (𝑎 · (𝑊‘𝐼)) = (𝑏 · (𝑊‘𝐼)))
99	98	eqeq2d 2750	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (((𝑃‘𝐼)‘𝑦) = (𝑎 · (𝑊‘𝐼)) ↔ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))
100	99	cbvrexvw 3218	. . . . . 6 ⊢ (∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑦) = (𝑎 · (𝑊‘𝐼)) ↔ ∃𝑏 ∈ ℤ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼)))
101	97, 100	bitrdi 288	. . . . 5 ⊢ (𝑣 = 𝑦 → (∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ∃𝑏 ∈ ℤ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))
102		simprr 778	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝑈)
103	101, 93, 102	rspcdva 3561	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ∃𝑏 ∈ ℤ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼)))
104		reeanv 3211	. . . . 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 732	. . . . . . . . 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 732	. . . . . . . . 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 784	. . . . . . . . 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 785	. . . . . . . . 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 14059	. . . . . . . . 9 ⊢ (((𝑇 ∈ ℂ ∧ 𝑇 ≠ 0) ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → (𝑇↑(𝑎 + 𝑏)) = ((𝑇↑𝑎) · (𝑇↑𝑏)))
110	105, 106, 107, 108, 109	syl22anc 844	. . . . . . . 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 772	. . . . . . . . 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 732	. . . . . . . . . 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 481	. . . . . . . . . 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 481	. . . . . . . . . 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 2739	. . . . . . . . . . 11 ⊢ (._r‘𝑍) = (._r‘𝑍)
116	4, 115	unitmulcl 20351	. . . . . . . . . 10 ⊢ ((𝑍 ∈ Ring ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈) → (𝑥(._r‘𝑍)𝑦) ∈ 𝑈)
117	112, 113, 114, 116	syl3anc 1379	. . . . . . . . 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 12628	. . . . . . . . 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 786	. . . . . . . . . . 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 787	. . . . . . . . . . 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 7374	. . . . . . . . . 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 20031	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃‘𝐼) ∈ ((𝐻 ↾_s (𝐻 DProd 𝑆)) GrpHom 𝐻))
123	21	oveq2d 7372	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐻 ↾_s (𝐻 DProd 𝑆)) = (𝐻 ↾_s 𝑈))
124	42	ovexi 7390	. . . . . . . . . . . . . . . 16 ⊢ 𝐻 ∈ V
125	69	ressid 17205	. . . . . . . . . . . . . . . 16 ⊢ (𝐻 ∈ V → (𝐻 ↾_s 𝑈) = 𝐻)
126	124, 125	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (𝐻 ↾_s 𝑈) = 𝐻
127	123, 126	eqtrdi 2790	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐻 ↾_s (𝐻 DProd 𝑆)) = 𝐻)
128	127	oveq1d 7371	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐻 ↾_s (𝐻 DProd 𝑆)) GrpHom 𝐻) = (𝐻 GrpHom 𝐻))
129	122, 128	eleqtrd 2841	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃‘𝐼) ∈ (𝐻 GrpHom 𝐻))
130	129	ad2antrr 732	. . . . . . . . . . 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 6841	. . . . . . . . . . . . 13 ⊢ 𝑈 ∈ V
132		eqid 2739	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
133	132, 115	mgpplusg 20116	. . . . . . . . . . . . . 14 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
134	42, 133	ressplusg 17245	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ V → (._r‘𝑍) = (+_g‘𝐻))
135	131, 134	ax-mp 5	. . . . . . . . . . . 12 ⊢ (._r‘𝑍) = (+_g‘𝐻)
136	69, 135, 135	ghmlin 19187	. . . . . . . . . . 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 1379	. . . . . . . . . 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 732	. . . . . . . . . . 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 732	. . . . . . . . . . 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 19073	. . . . . . . . . . 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 1380	. . . . . . . . . 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 2784	. . . . . . . . 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 27245	. . . . . . . . 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 844	. . . . . . . 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 27245	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)))) → (𝑋‘𝑥) = (𝑇↑𝑎))
146	111, 113, 107, 119, 145	syl22anc 844	. . . . . . . . 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 27245	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑈) ∧ (𝑏 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼)))) → (𝑋‘𝑦) = (𝑇↑𝑏))
148	111, 114, 108, 120, 147	syl22anc 844	. . . . . . . . 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 7374	. . . . . . . 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 2784	. . . . . . 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 457	. . . . . 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 3196	. . . . 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 244	. . . 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 705	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑋‘(𝑥(._r‘𝑍)𝑦)) = ((𝑋‘𝑥) · (𝑋‘𝑦)))
155		id 22	. . . . 5 ⊢ (𝜑 → 𝜑)
156		eqid 2739	. . . . . . 7 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
157	4, 156	1unit 20345	. . . . . 6 ⊢ (𝑍 ∈ Ring → (1_r‘𝑍) ∈ 𝑈)
158	56, 157	syl 17	. . . . 5 ⊢ (𝜑 → (1_r‘𝑍) ∈ 𝑈)
159		0zd 12527	. . . . 5 ⊢ (𝜑 → 0 ∈ ℤ)
160		eqid 2739	. . . . . . . 8 ⊢ (0_g‘𝐻) = (0_g‘𝐻)
161	160, 160	ghmid 19188	. . . . . . 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 20356	. . . . . . . 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 6831	. . . . . 6 ⊢ (𝜑 → ((𝑃‘𝐼)‘(1_r‘𝑍)) = ((𝑃‘𝐼)‘(0_g‘𝐻)))
166	69, 160, 43	mulg0 19041	. . . . . . 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 2784	. . . . 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 27245	. . . . 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 844	. . . 4 ⊢ (𝜑 → (𝑋‘(1_r‘𝑍)) = (𝑇↑0))
171	77	exp0d 14093	. . . 4 ⊢ (𝜑 → (𝑇↑0) = 1)
172	170, 171	eqtrd 2774	. . 3 ⊢ (𝜑 → (𝑋‘(1_r‘𝑍)) = 1)
173	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 89, 154, 172	dchrelbasd 27220	. 2 ⊢ (𝜑 → (𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0)) ∈ 𝐷)
174	61, 44	sselid 3913	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
175		eleq1 2827	. . . . . . 7 ⊢ (𝑣 = 𝐴 → (𝑣 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈))
176		fveq2 6827	. . . . . . 7 ⊢ (𝑣 = 𝐴 → (𝑋‘𝑣) = (𝑋‘𝐴))
177	175, 176	ifbieq1d 4479	. . . . . 6 ⊢ (𝑣 = 𝐴 → if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0) = if(𝐴 ∈ 𝑈, (𝑋‘𝐴), 0))
178		eqid 2739	. . . . . 6 ⊢ (𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0)) = (𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0))
179		fvex 6840	. . . . . . 7 ⊢ (𝑋‘𝑣) ∈ V
180		c0ex 11129	. . . . . . 7 ⊢ 0 ∈ V
181	179, 180	ifex 4505	. . . . . 6 ⊢ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0) ∈ V
182	177, 178, 181	fvmpt3i 6941	. . . . 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 4462	. . . 4 ⊢ (𝜑 → if(𝐴 ∈ 𝑈, (𝑋‘𝐴), 0) = (𝑋‘𝐴))
185	183, 184	eqtrd 2774	. . 3 ⊢ (𝜑 → ((𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0))‘𝐴) = (𝑋‘𝐴))
186		fveqeq2 6836	. . . . . 6 ⊢ (𝑣 = 𝐴 → (((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼))))
187	186	rexbidv 3163	. . . . 5 ⊢ (𝑣 = 𝐴 → (∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼))))
188	187, 92, 44	rspcdva 3561	. . . 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 27245	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → (𝑋‘𝐴) = (𝑇↑𝑎))
190	47	oveq1i 7366	. . . . . . . 8 ⊢ (𝑇↑𝑎) = ((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑𝑎)
191	189, 190	eqtrdi 2790	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → (𝑋‘𝐴) = ((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑𝑎))
192	48	ad2antrr 732	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → ((𝑃‘𝐼)‘𝐴) ≠ 1 )
193	58	ad2antrr 732	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → 𝐻 ∈ Grp)
194	68	ad2antrr 732	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → (𝑊‘𝐼) ∈ 𝑈)
195		simprl 776	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → 𝑎 ∈ ℤ)
196	69, 46, 43, 160	oddvds 19513	. . . . . . . . . . 11 ⊢ ((𝐻 ∈ Grp ∧ (𝑊‘𝐼) ∈ 𝑈 ∧ 𝑎 ∈ ℤ) → ((𝑂‘(𝑊‘𝐼)) ∥ 𝑎 ↔ (𝑎 · (𝑊‘𝐼)) = (0_g‘𝐻)))
197	193, 194, 195, 196	syl3anc 1379	. . . . . . . . . 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 732	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → (𝑂‘(𝑊‘𝐼)) ∈ ℕ)
199		root1eq1 26737	. . . . . . . . . . 11 ⊢ (((𝑂‘(𝑊‘𝐼)) ∈ ℕ ∧ 𝑎 ∈ ℤ) → (((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑𝑎) = 1 ↔ (𝑂‘(𝑊‘𝐼)) ∥ 𝑎))
200	198, 195, 199	syl2anc 590	. . . . . . . . . 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 778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))
202	40, 164	eqtrid 2786	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 = (0_g‘𝐻))
203	202	ad2antrr 732	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → 1 = (0_g‘𝐻))
204	201, 203	eqeq12d 2755	. . . . . . . . . 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 312	. . . . . . . . 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 2978	. . . . . . . 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 258	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → ((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑𝑎) ≠ 1)
208	191, 207	eqnetrd 3001	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → (𝑋‘𝐴) ≠ 1)
209	208	rexlimdvaa 3141	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)) → (𝑋‘𝐴) ≠ 1))
210	44, 209	mpdan 693	. . . 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 3001	. 2 ⊢ (𝜑 → ((𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0))‘𝐴) ≠ 1)
213		fveq1 6826	. . . 4 ⊢ (𝑥 = (𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0)) → (𝑥‘𝐴) = ((𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0))‘𝐴))
214	213	neeq1d 2993	. . 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 3560	. 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 590	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∀wral 3053 ∃wrex 3063 Vcvv 3431 ⊆ wss 3883 ifcif 4454 class class class wbr 5072 ↦ cmpt 5153 dom cdm 5618 ran crn 5619 ℩cio 6439 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 Fincfn 8883 ℂcc 11027 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 · cmul 11034 -cneg 11369 / cdiv 11798 ℕcn 12165 2c2 12227 ℕ₀cn0 12428 ℤcz 12515 ..^cfzo 13599 ↑cexp 14014 ♯chash 14283 Word cword 14466 ∥ cdvds 16212 Basecbs 17170 ↾_s cress 17191 +_gcplusg 17211 ._rcmulr 17212 0_gc0g 17393 Grpcgrp 18900 ._gcmg 19034 GrpHom cghm 19178 odcod 19490 DProd cdprd 19961 dProjcdpj 19962 mulGrpcmgp 20112 1_rcur 20153 Ringcrg 20205 CRingccrg 20206 Unitcui 20326 ℤ/nℤczn 21477 ↑_𝑐ccxp 26537 DChrcdchr 27213

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108 ax-mulf 11109

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-omul 8400 df-er 8633 df-ec 8635 df-qs 8639 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9854 df-acn 9857 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ioc 13294 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-fac 14227 df-bc 14256 df-hash 14284 df-word 14467 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15424 df-clim 15441 df-rlim 15442 df-sum 15640 df-ef 16023 df-sin 16025 df-cos 16026 df-pi 16028 df-dvds 16213 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-qus 17464 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18742 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-mulg 19035 df-subg 19090 df-nsg 19091 df-eqg 19092 df-ghm 19179 df-gim 19225 df-cntz 19283 df-oppg 19312 df-od 19494 df-lsm 19602 df-pj1 19603 df-cmn 19748 df-abl 19749 df-dprd 19963 df-dpj 19964 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-cring 20208 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-rhm 20443 df-subrng 20518 df-subrg 20542 df-lmod 20852 df-lss 20922 df-lsp 20962 df-sra 21163 df-rgmod 21164 df-lidl 21201 df-rsp 21202 df-2idl 21243 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-zring 21422 df-zrh 21478 df-zn 21481 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-cld 23002 df-ntr 23003 df-cls 23004 df-nei 23081 df-lp 23119 df-perf 23120 df-cn 23210 df-cnp 23211 df-haus 23298 df-tx 23545 df-hmeo 23738 df-fil 23829 df-fm 23921 df-flim 23922 df-flf 23923 df-xms 24303 df-ms 24304 df-tms 24305 df-cncf 24863 df-limc 25851 df-dv 25852 df-log 26538 df-cxp 26539 df-dchr 27214

This theorem is referenced by: dchrptlem3 27247

W3C validator