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Theorem dchrptlem2 27440

Description: Lemma for dchrpt 27442. (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 6881	. . 3 ⊢ (𝑣 = 𝑥 → (𝑋‘𝑣) = (𝑋‘𝑥))
8		fveq2 6881	. . 3 ⊢ (𝑣 = 𝑦 → (𝑋‘𝑣) = (𝑋‘𝑦))
9		fveq2 6881	. . 3 ⊢ (𝑣 = (𝑥(._r‘𝑍)𝑦) → (𝑋‘𝑣) = (𝑋‘(𝑥(._r‘𝑍)𝑦)))
10		fveq2 6881	. . 3 ⊢ (𝑣 = (1_r‘𝑍) → (𝑋‘𝑣) = (𝑋‘(1_r‘𝑍)))
11		dchrpt.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐻dom DProd 𝑆)
12		zex 12606	. . . . . . . . . . . . 13 ⊢ ℤ ∈ V
13	12	mptex 7221	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) ∈ V
14	13	rnex 7905	. . . . . . . . . . 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 6679	. . . . . . . . . 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 20135	. . . . . . . 8 ⊢ (𝜑 → (𝑃‘𝐼):(𝐻 DProd 𝑆)⟶(𝑆‘𝐼))
21		dchrpt.3	. . . . . . . . 9 ⊢ (𝜑 → (𝐻 DProd 𝑆) = 𝑈)
22	21	feq2d 6689	. . . . . . . 8 ⊢ (𝜑 → ((𝑃‘𝐼):(𝐻 DProd 𝑆)⟶(𝑆‘𝐼) ↔ (𝑃‘𝐼):𝑈⟶(𝑆‘𝐼)))
23	20, 22	mpbid 235	. . . . . . 7 ⊢ (𝜑 → (𝑃‘𝐼):𝑈⟶(𝑆‘𝐼))
24	23	ffvelcdmda 7079	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑈) → ((𝑃‘𝐼)‘𝑣) ∈ (𝑆‘𝐼))
25	19	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑈) → 𝐼 ∈ dom 𝑊)
26		oveq1 7419	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑎 → (𝑛 · (𝑊‘𝑘)) = (𝑎 · (𝑊‘𝑘)))
27	26	cbvmptv 5214	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) = (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝑘)))
28		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐼 → (𝑊‘𝑘) = (𝑊‘𝐼))
29	28	oveq2d 7428	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐼 → (𝑎 · (𝑊‘𝑘)) = (𝑎 · (𝑊‘𝐼)))
30	29	mpteq2dv 5204	. . . . . . . . . 10 ⊢ (𝑘 = 𝐼 → (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝑘))) = (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))))
31	27, 30	eqtrid 2809	. . . . . . . . 9 ⊢ (𝑘 = 𝐼 → (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) = (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))))
32	31	rneqd 5927	. . . . . . . 8 ⊢ (𝑘 = 𝐼 → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) = ran (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))))
33	32, 15, 14	fvmpt3i 6995	. . . . . . 7 ⊢ (𝐼 ∈ dom 𝑊 → (𝑆‘𝐼) = ran (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))))
34	25, 33	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑈) → (𝑆‘𝐼) = ran (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))))
35	24, 34	eleqtrd 2864	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑈) → ((𝑃‘𝐼)‘𝑣) ∈ ran (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))))
36		eqid 2762	. . . . . 6 ⊢ (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))) = (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼)))
37		ovex 7445	. . . . . 6 ⊢ (𝑎 · (𝑊‘𝐼)) ∈ V
38	36, 37	elrnmpti 5951	. . . . 5 ⊢ (((𝑃‘𝐼)‘𝑣) ∈ ran (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))) ↔ ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))
39	35, 38	sylib 221	. . . 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 27439	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))) → (𝑋‘𝑣) = (𝑇↑𝑎))
51		neg1cn 12209	. . . . . . . . 9 ⊢ -1 ∈ ℂ
52		2re 12321	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
53	5	nnnn0d 12571	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
54	2	zncrng 21705	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
55		crngring 20333	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
56	53, 54, 55	3syl 19	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ Ring)
57	4, 42	unitgrp 20472	. . . . . . . . . . . . 13 ⊢ (𝑍 ∈ Ring → 𝐻 ∈ Grp)
58	56, 57	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ∈ Grp)
59	2, 3	znfi 21720	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 𝐵 ∈ Fin)
60	5, 59	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ Fin)
61	3, 4	unitss 20465	. . . . . . . . . . . . 13 ⊢ 𝑈 ⊆ 𝐵
62		ssfi 9155	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ Fin ∧ 𝑈 ⊆ 𝐵) → 𝑈 ∈ Fin)
63	60, 61, 62	sylancl 597	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ Fin)
64		wrdf 14562	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ Word 𝑈 → 𝑊:(0..^(♯‘𝑊))⟶𝑈)
65	45, 64	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑊:(0..^(♯‘𝑊))⟶𝑈)
66	65	fdmd 6716	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
67	19, 66	eleqtrd 2864	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ (0..^(♯‘𝑊)))
68	65, 67	ffvelcdmd 7080	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑊‘𝐼) ∈ 𝑈)
69	4, 42	unitgrpbas 20471	. . . . . . . . . . . . 13 ⊢ 𝑈 = (Base‘𝐻)
70	69, 46	odcl2 19641	. . . . . . . . . . . 12 ⊢ ((𝐻 ∈ Grp ∧ 𝑈 ∈ Fin ∧ (𝑊‘𝐼) ∈ 𝑈) → (𝑂‘(𝑊‘𝐼)) ∈ ℕ)
71	58, 63, 68, 70	syl3anc 1397	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂‘(𝑊‘𝐼)) ∈ ℕ)
72		nndivre 12283	. . . . . . . . . . 11 ⊢ ((2 ∈ ℝ ∧ (𝑂‘(𝑊‘𝐼)) ∈ ℕ) → (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℝ)
73	52, 71, 72	sylancr 598	. . . . . . . . . 10 ⊢ (𝜑 → (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℝ)
74	73	recnd 11243	. . . . . . . . 9 ⊢ (𝜑 → (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℂ)
75		cxpcl 26850	. . . . . . . . 9 ⊢ ((-1 ∈ ℂ ∧ (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℂ) → (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ∈ ℂ)
76	51, 74, 75	sylancr 598	. . . . . . . 8 ⊢ (𝜑 → (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ∈ ℂ)
77	47, 76	eqeltrid 2866	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ ℂ)
78	77	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))) → 𝑇 ∈ ℂ)
79	51	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → -1 ∈ ℂ)
80		neg1ne0 12211	. . . . . . . . . 10 ⊢ -1 ≠ 0
81	80	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → -1 ≠ 0)
82	79, 81, 74	cxpne0d 26889	. . . . . . . 8 ⊢ (𝜑 → (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ≠ 0)
83	47	neeq1i 3021	. . . . . . . 8 ⊢ (𝑇 ≠ 0 ↔ (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ≠ 0)
84	82, 83	sylibr 237	. . . . . . 7 ⊢ (𝜑 → 𝑇 ≠ 0)
85	84	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))) → 𝑇 ≠ 0)
86		simprl 782	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))) → 𝑎 ∈ ℤ)
87	78, 85, 86	expclzd 14194	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))) → (𝑇↑𝑎) ∈ ℂ)
88	50, 87	eqeltrd 2862	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))) → (𝑋‘𝑣) ∈ ℂ)
89	39, 88	rexlimddv 3171	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑈) → (𝑋‘𝑣) ∈ ℂ)
90		fveqeq2 6890	. . . . . 6 ⊢ (𝑣 = 𝑥 → (((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼))))
91	90	rexbidv 3188	. . . . 5 ⊢ (𝑣 = 𝑥 → (∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼))))
92	39	ralrimiva 3156	. . . . . 6 ⊢ (𝜑 → ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))
93	92	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))
94		simprl 782	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝑈)
95	91, 93, 94	rspcdva 3581	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)))
96		fveqeq2 6890	. . . . . . 7 ⊢ (𝑣 = 𝑦 → (((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ((𝑃‘𝐼)‘𝑦) = (𝑎 · (𝑊‘𝐼))))
97	96	rexbidv 3188	. . . . . 6 ⊢ (𝑣 = 𝑦 → (∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑦) = (𝑎 · (𝑊‘𝐼))))
98		oveq1 7419	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (𝑎 · (𝑊‘𝐼)) = (𝑏 · (𝑊‘𝐼)))
99	98	eqeq2d 2773	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (((𝑃‘𝐼)‘𝑦) = (𝑎 · (𝑊‘𝐼)) ↔ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))
100	99	cbvrexvw 3243	. . . . . 6 ⊢ (∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑦) = (𝑎 · (𝑊‘𝐼)) ↔ ∃𝑏 ∈ ℤ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼)))
101	97, 100	bitrdi 290	. . . . 5 ⊢ (𝑣 = 𝑦 → (∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ∃𝑏 ∈ ℤ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))
102		simprr 784	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝑈)
103	101, 93, 102	rspcdva 3581	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ∃𝑏 ∈ ℤ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼)))
104		reeanv 3236	. . . . 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 738	. . . . . . . . 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 738	. . . . . . . . 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 790	. . . . . . . . 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 791	. . . . . . . . 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 14149	. . . . . . . . 9 ⊢ (((𝑇 ∈ ℂ ∧ 𝑇 ≠ 0) ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ)) → (𝑇↑(𝑎 + 𝑏)) = ((𝑇↑𝑎) · (𝑇↑𝑏)))
110	105, 106, 107, 108, 109	syl22anc 851	. . . . . . . 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 778	. . . . . . . . 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 738	. . . . . . . . . 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 485	. . . . . . . . . 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 485	. . . . . . . . . 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 2762	. . . . . . . . . . 11 ⊢ (._r‘𝑍) = (._r‘𝑍)
116	4, 115	unitmulcl 20469	. . . . . . . . . 10 ⊢ ((𝑍 ∈ Ring ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈) → (𝑥(._r‘𝑍)𝑦) ∈ 𝑈)
117	112, 113, 114, 116	syl3anc 1397	. . . . . . . . 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 12710	. . . . . . . . 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 792	. . . . . . . . . . 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 793	. . . . . . . . . . 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 7430	. . . . . . . . . 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 20141	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃‘𝐼) ∈ ((𝐻 ↾_s (𝐻 DProd 𝑆)) GrpHom 𝐻))
123	21	oveq2d 7428	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐻 ↾_s (𝐻 DProd 𝑆)) = (𝐻 ↾_s 𝑈))
124	42	ovexi 7446	. . . . . . . . . . . . . . . 16 ⊢ 𝐻 ∈ V
125	69	ressid 17310	. . . . . . . . . . . . . . . 16 ⊢ (𝐻 ∈ V → (𝐻 ↾_s 𝑈) = 𝐻)
126	124, 125	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (𝐻 ↾_s 𝑈) = 𝐻
127	123, 126	eqtrdi 2813	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐻 ↾_s (𝐻 DProd 𝑆)) = 𝐻)
128	127	oveq1d 7427	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐻 ↾_s (𝐻 DProd 𝑆)) GrpHom 𝐻) = (𝐻 GrpHom 𝐻))
129	122, 128	eleqtrd 2864	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃‘𝐼) ∈ (𝐻 GrpHom 𝐻))
130	129	ad2antrr 738	. . . . . . . . . . 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 6895	. . . . . . . . . . . . 13 ⊢ 𝑈 ∈ V
132		eqid 2762	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
133	132, 115	mgpplusg 20226	. . . . . . . . . . . . . 14 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
134	42, 133	ressplusg 17350	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ V → (._r‘𝑍) = (+_g‘𝐻))
135	131, 134	ax-mp 5	. . . . . . . . . . . 12 ⊢ (._r‘𝑍) = (+_g‘𝐻)
136	69, 135, 135	ghmlin 19297	. . . . . . . . . . 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 1397	. . . . . . . . . 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 738	. . . . . . . . . . 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 738	. . . . . . . . . . 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 19178	. . . . . . . . . . 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 1398	. . . . . . . . . 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 2807	. . . . . . . . 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 27439	. . . . . . . . 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 851	. . . . . . . 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 27439	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)))) → (𝑋‘𝑥) = (𝑇↑𝑎))
146	111, 113, 107, 119, 145	syl22anc 851	. . . . . . . . 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 27439	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑈) ∧ (𝑏 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼)))) → (𝑋‘𝑦) = (𝑇↑𝑏))
148	111, 114, 108, 120, 147	syl22anc 851	. . . . . . . . 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 7430	. . . . . . . 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 2807	. . . . . . 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 461	. . . . . 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 3221	. . . . 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 246	. . . 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 711	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑋‘(𝑥(._r‘𝑍)𝑦)) = ((𝑋‘𝑥) · (𝑋‘𝑦)))
155		id 23	. . . . 5 ⊢ (𝜑 → 𝜑)
156		eqid 2762	. . . . . . 7 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
157	4, 156	1unit 20463	. . . . . 6 ⊢ (𝑍 ∈ Ring → (1_r‘𝑍) ∈ 𝑈)
158	56, 157	syl 18	. . . . 5 ⊢ (𝜑 → (1_r‘𝑍) ∈ 𝑈)
159		0zd 12609	. . . . 5 ⊢ (𝜑 → 0 ∈ ℤ)
160		eqid 2762	. . . . . . . 8 ⊢ (0_g‘𝐻) = (0_g‘𝐻)
161	160, 160	ghmid 19298	. . . . . . 7 ⊢ ((𝑃‘𝐼) ∈ (𝐻 GrpHom 𝐻) → ((𝑃‘𝐼)‘(0_g‘𝐻)) = (0_g‘𝐻))
162	129, 161	syl 18	. . . . . 6 ⊢ (𝜑 → ((𝑃‘𝐼)‘(0_g‘𝐻)) = (0_g‘𝐻))
163	4, 42, 156	unitgrpid 20474	. . . . . . . 8 ⊢ (𝑍 ∈ Ring → (1_r‘𝑍) = (0_g‘𝐻))
164	56, 163	syl 18	. . . . . . 7 ⊢ (𝜑 → (1_r‘𝑍) = (0_g‘𝐻))
165	164	fveq2d 6885	. . . . . 6 ⊢ (𝜑 → ((𝑃‘𝐼)‘(1_r‘𝑍)) = ((𝑃‘𝐼)‘(0_g‘𝐻)))
166	69, 160, 43	mulg0 19146	. . . . . . 7 ⊢ ((𝑊‘𝐼) ∈ 𝑈 → (0 · (𝑊‘𝐼)) = (0_g‘𝐻))
167	68, 166	syl 18	. . . . . 6 ⊢ (𝜑 → (0 · (𝑊‘𝐼)) = (0_g‘𝐻))
168	162, 165, 167	3eqtr4d 2807	. . . . 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 27439	. . . . 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 851	. . . 4 ⊢ (𝜑 → (𝑋‘(1_r‘𝑍)) = (𝑇↑0))
171	77	exp0d 14183	. . . 4 ⊢ (𝜑 → (𝑇↑0) = 1)
172	170, 171	eqtrd 2797	. . 3 ⊢ (𝜑 → (𝑋‘(1_r‘𝑍)) = 1)
173	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 89, 154, 172	dchrelbasd 27414	. 2 ⊢ (𝜑 → (𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0)) ∈ 𝐷)
174	61, 44	sselid 3934	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
175		eleq1 2850	. . . . . . 7 ⊢ (𝑣 = 𝐴 → (𝑣 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈))
176		fveq2 6881	. . . . . . 7 ⊢ (𝑣 = 𝐴 → (𝑋‘𝑣) = (𝑋‘𝐴))
177	175, 176	ifbieq1d 4511	. . . . . 6 ⊢ (𝑣 = 𝐴 → if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0) = if(𝐴 ∈ 𝑈, (𝑋‘𝐴), 0))
178		eqid 2762	. . . . . 6 ⊢ (𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0)) = (𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0))
179		fvex 6894	. . . . . . 7 ⊢ (𝑋‘𝑣) ∈ V
180		c0ex 11206	. . . . . . 7 ⊢ 0 ∈ V
181	179, 180	ifex 4537	. . . . . 6 ⊢ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0) ∈ V
182	177, 178, 181	fvmpt3i 6995	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → ((𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0))‘𝐴) = if(𝐴 ∈ 𝑈, (𝑋‘𝐴), 0))
183	174, 182	syl 18	. . . 4 ⊢ (𝜑 → ((𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0))‘𝐴) = if(𝐴 ∈ 𝑈, (𝑋‘𝐴), 0))
184	44	iftrued 4494	. . . 4 ⊢ (𝜑 → if(𝐴 ∈ 𝑈, (𝑋‘𝐴), 0) = (𝑋‘𝐴))
185	183, 184	eqtrd 2797	. . 3 ⊢ (𝜑 → ((𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0))‘𝐴) = (𝑋‘𝐴))
186		fveqeq2 6890	. . . . . 6 ⊢ (𝑣 = 𝐴 → (((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼))))
187	186	rexbidv 3188	. . . . 5 ⊢ (𝑣 = 𝐴 → (∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼))))
188	187, 92, 44	rspcdva 3581	. . . 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 27439	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → (𝑋‘𝐴) = (𝑇↑𝑎))
190	47	oveq1i 7422	. . . . . . . 8 ⊢ (𝑇↑𝑎) = ((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑𝑎)
191	189, 190	eqtrdi 2813	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → (𝑋‘𝐴) = ((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑𝑎))
192	48	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → ((𝑃‘𝐼)‘𝐴) ≠ 1 )
193	58	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → 𝐻 ∈ Grp)
194	68	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → (𝑊‘𝐼) ∈ 𝑈)
195		simprl 782	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → 𝑎 ∈ ℤ)
196	69, 46, 43, 160	oddvds 19623	. . . . . . . . . . 11 ⊢ ((𝐻 ∈ Grp ∧ (𝑊‘𝐼) ∈ 𝑈 ∧ 𝑎 ∈ ℤ) → ((𝑂‘(𝑊‘𝐼)) ∥ 𝑎 ↔ (𝑎 · (𝑊‘𝐼)) = (0_g‘𝐻)))
197	193, 194, 195, 196	syl3anc 1397	. . . . . . . . . 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 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → (𝑂‘(𝑊‘𝐼)) ∈ ℕ)
199		root1eq1 26931	. . . . . . . . . . 11 ⊢ (((𝑂‘(𝑊‘𝐼)) ∈ ℕ ∧ 𝑎 ∈ ℤ) → (((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑𝑎) = 1 ↔ (𝑂‘(𝑊‘𝐼)) ∥ 𝑎))
200	198, 195, 199	syl2anc 595	. . . . . . . . . 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 784	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))
202	40, 164	eqtrid 2809	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 = (0_g‘𝐻))
203	202	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → 1 = (0_g‘𝐻))
204	201, 203	eqeq12d 2778	. . . . . . . . . 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 314	. . . . . . . . 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 3001	. . . . . . . 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 260	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → ((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑𝑎) ≠ 1)
208	191, 207	eqnetrd 3024	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → (𝑋‘𝐴) ≠ 1)
209	208	rexlimdvaa 3166	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)) → (𝑋‘𝐴) ≠ 1))
210	44, 209	mpdan 699	. . . 4 ⊢ (𝜑 → (∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)) → (𝑋‘𝐴) ≠ 1))
211	188, 210	mpd 16	. . 3 ⊢ (𝜑 → (𝑋‘𝐴) ≠ 1)
212	185, 211	eqnetrd 3024	. 2 ⊢ (𝜑 → ((𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0))‘𝐴) ≠ 1)
213		fveq1 6880	. . . 4 ⊢ (𝑥 = (𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0)) → (𝑥‘𝐴) = ((𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0))‘𝐴))
214	213	neeq1d 3016	. . 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 3580	. 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 595	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1569 ∈ wcel 2142 ≠ wne 2957 ∀wral 3078 ∃wrex 3088 Vcvv 3454 ⊆ wss 3904 ifcif 4486 class class class wbr 5108 ↦ cmpt 5191 dom cdm 5660 ran crn 5661 ℩cio 6490 ⟶wf 6532 ‘cfv 6536 (class class class)co 7412 Fincfn 8941 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 -cneg 11448 / cdiv 11877 ℕcn 12239 2c2 12301 ℕ₀cn0 12510 ℤcz 12597 ..^cfzo 13689 ↑cexp 14104 ♯chash 14373 Word cword 14557 ∥ cdvds 16316 Basecbs 17275 ↾_s cress 17296 +_gcplusg 17316 ._rcmulr 17317 0_gc0g 17498 Grpcgrp 19006 ._gcmg 19139 GrpHom cghm 19289 odcod 19600 DProd cdprd 20071 dProjcdpj 20072 mulGrpcmgp 20222 1_rcur 20269 Ringcrg 20321 CRingccrg 20322 Unitcui 20444 ℤ/nℤczn 21663 ↑_𝑐ccxp 26731 DChrcdchr 27407

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-inf2 9608 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8155 df-tpos 8220 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-oadd 8455 df-omul 8456 df-er 8692 df-ec 8694 df-qs 8698 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9932 df-acn 9935 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13143 df-xadd 13144 df-xmul 13145 df-ioo 13382 df-ioc 13383 df-ico 13384 df-icc 13385 df-fz 13542 df-fzo 13690 df-fl 13832 df-mod 13910 df-seq 14045 df-exp 14105 df-fac 14317 df-bc 14346 df-hash 14374 df-word 14558 df-shft 15111 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-limsup 15529 df-clim 15546 df-rlim 15547 df-sum 15745 df-ef 16127 df-sin 16129 df-cos 16130 df-pi 16132 df-dvds 16317 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-starv 17331 df-sca 17332 df-vsca 17333 df-ip 17334 df-tset 17335 df-ple 17336 df-ds 17338 df-unif 17339 df-hom 17340 df-cco 17341 df-rest 17481 df-topn 17482 df-0g 17500 df-gsum 17501 df-topgen 17502 df-pt 17503 df-prds 17506 df-xrs 17562 df-qtop 17567 df-imas 17568 df-qus 17569 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-mhm 18847 df-submnd 18848 df-grp 19009 df-minusg 19010 df-sbg 19011 df-mulg 19140 df-subg 19195 df-nsg 19196 df-eqg 19197 df-ghm 19290 df-gim 19335 df-cntz 19393 df-oppg 19422 df-od 19604 df-lsm 19712 df-pj1 19713 df-cmn 19858 df-abl 19859 df-dprd 20073 df-dpj 20074 df-mgp 20223 df-rng 20237 df-ur 20270 df-ring 20323 df-cring 20324 df-oppr 20426 df-dvdsr 20446 df-unit 20447 df-rhm 20561 df-subrng 20656 df-subrg 20680 df-lmod 20994 df-lss 21064 df-lsp 21104 df-sra 21305 df-rgmod 21306 df-lidl 21343 df-rsp 21344 df-2idl 21400 df-psmet 21525 df-xmet 21526 df-met 21527 df-bl 21528 df-mopn 21529 df-fbas 21530 df-fg 21531 df-cnfld 21534 df-zring 21608 df-zrh 21664 df-zn 21667 df-top 23062 df-topon 23079 df-topsp 23101 df-bases 23114 df-cld 23187 df-ntr 23188 df-cls 23189 df-nei 23266 df-lp 23304 df-perf 23305 df-cn 23395 df-cnp 23396 df-haus 23483 df-tx 23730 df-hmeo 23923 df-fil 24014 df-fm 24106 df-flim 24107 df-flf 24108 df-xms 24488 df-ms 24489 df-tms 24490 df-cncf 25048 df-limc 26036 df-dv 26037 df-log 26732 df-cxp 26733 df-dchr 27408

This theorem is used by: dchrptlem3 27441

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