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Theorem dchrptlem2 27176

Description: Lemma for dchrpt 27178. (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 6858	. . 3 ⊢ (𝑣 = 𝑥 → (𝑋‘𝑣) = (𝑋‘𝑥))
8		fveq2 6858	. . 3 ⊢ (𝑣 = 𝑦 → (𝑋‘𝑣) = (𝑋‘𝑦))
9		fveq2 6858	. . 3 ⊢ (𝑣 = (𝑥(._r‘𝑍)𝑦) → (𝑋‘𝑣) = (𝑋‘(𝑥(._r‘𝑍)𝑦)))
10		fveq2 6858	. . 3 ⊢ (𝑣 = (1_r‘𝑍) → (𝑋‘𝑣) = (𝑋‘(1_r‘𝑍)))
11		dchrpt.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐻dom DProd 𝑆)
12		zex 12538	. . . . . . . . . . . . 13 ⊢ ℤ ∈ V
13	12	mptex 7197	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) ∈ V
14	13	rnex 7886	. . . . . . . . . . 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 6662	. . . . . . . . . 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 19989	. . . . . . . 8 ⊢ (𝜑 → (𝑃‘𝐼):(𝐻 DProd 𝑆)⟶(𝑆‘𝐼))
21		dchrpt.3	. . . . . . . . 9 ⊢ (𝜑 → (𝐻 DProd 𝑆) = 𝑈)
22	21	feq2d 6672	. . . . . . . 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 7056	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑈) → ((𝑃‘𝐼)‘𝑣) ∈ (𝑆‘𝐼))
25	19	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑈) → 𝐼 ∈ dom 𝑊)
26		oveq1 7394	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑎 → (𝑛 · (𝑊‘𝑘)) = (𝑎 · (𝑊‘𝑘)))
27	26	cbvmptv 5211	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) = (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝑘)))
28		fveq2 6858	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐼 → (𝑊‘𝑘) = (𝑊‘𝐼))
29	28	oveq2d 7403	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐼 → (𝑎 · (𝑊‘𝑘)) = (𝑎 · (𝑊‘𝐼)))
30	29	mpteq2dv 5201	. . . . . . . . . 10 ⊢ (𝑘 = 𝐼 → (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝑘))) = (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))))
31	27, 30	eqtrid 2776	. . . . . . . . 9 ⊢ (𝑘 = 𝐼 → (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) = (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))))
32	31	rneqd 5902	. . . . . . . 8 ⊢ (𝑘 = 𝐼 → ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))) = ran (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))))
33	32, 15, 14	fvmpt3i 6973	. . . . . . 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 2830	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑈) → ((𝑃‘𝐼)‘𝑣) ∈ ran (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))))
36		eqid 2729	. . . . . 6 ⊢ (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼))) = (𝑎 ∈ ℤ ↦ (𝑎 · (𝑊‘𝐼)))
37		ovex 7420	. . . . . 6 ⊢ (𝑎 · (𝑊‘𝐼)) ∈ V
38	36, 37	elrnmpti 5926	. . . . 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 27175	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))) → (𝑋‘𝑣) = (𝑇↑𝑎))
51		neg1cn 12171	. . . . . . . . 9 ⊢ -1 ∈ ℂ
52		2re 12260	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
53	5	nnnn0d 12503	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
54	2	zncrng 21454	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
55		crngring 20154	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
56	53, 54, 55	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ Ring)
57	4, 42	unitgrp 20292	. . . . . . . . . . . . 13 ⊢ (𝑍 ∈ Ring → 𝐻 ∈ Grp)
58	56, 57	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ∈ Grp)
59	2, 3	znfi 21469	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 𝐵 ∈ Fin)
60	5, 59	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ Fin)
61	3, 4	unitss 20285	. . . . . . . . . . . . 13 ⊢ 𝑈 ⊆ 𝐵
62		ssfi 9137	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ Fin ∧ 𝑈 ⊆ 𝐵) → 𝑈 ∈ Fin)
63	60, 61, 62	sylancl 586	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ Fin)
64		wrdf 14483	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ Word 𝑈 → 𝑊:(0..^(♯‘𝑊))⟶𝑈)
65	45, 64	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑊:(0..^(♯‘𝑊))⟶𝑈)
66	65	fdmd 6698	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
67	19, 66	eleqtrd 2830	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ (0..^(♯‘𝑊)))
68	65, 67	ffvelcdmd 7057	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑊‘𝐼) ∈ 𝑈)
69	4, 42	unitgrpbas 20291	. . . . . . . . . . . . 13 ⊢ 𝑈 = (Base‘𝐻)
70	69, 46	odcl2 19495	. . . . . . . . . . . 12 ⊢ ((𝐻 ∈ Grp ∧ 𝑈 ∈ Fin ∧ (𝑊‘𝐼) ∈ 𝑈) → (𝑂‘(𝑊‘𝐼)) ∈ ℕ)
71	58, 63, 68, 70	syl3anc 1373	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂‘(𝑊‘𝐼)) ∈ ℕ)
72		nndivre 12227	. . . . . . . . . . 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 11202	. . . . . . . . 9 ⊢ (𝜑 → (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℂ)
75		cxpcl 26583	. . . . . . . . 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 2832	. . . . . . 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 12173	. . . . . . . . . 10 ⊢ -1 ≠ 0
81	80	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → -1 ≠ 0)
82	79, 81, 74	cxpne0d 26622	. . . . . . . 8 ⊢ (𝜑 → (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ≠ 0)
83	47	neeq1i 2989	. . . . . . . 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 14116	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))) → (𝑇↑𝑎) ∈ ℂ)
88	50, 87	eqeltrd 2828	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)))) → (𝑋‘𝑣) ∈ ℂ)
89	39, 88	rexlimddv 3140	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑈) → (𝑋‘𝑣) ∈ ℂ)
90		fveqeq2 6867	. . . . . 6 ⊢ (𝑣 = 𝑥 → (((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼))))
91	90	rexbidv 3157	. . . . 5 ⊢ (𝑣 = 𝑥 → (∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼))))
92	39	ralrimiva 3125	. . . . . 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 3589	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑥) = (𝑎 · (𝑊‘𝐼)))
96		fveqeq2 6867	. . . . . . 7 ⊢ (𝑣 = 𝑦 → (((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ((𝑃‘𝐼)‘𝑦) = (𝑎 · (𝑊‘𝐼))))
97	96	rexbidv 3157	. . . . . 6 ⊢ (𝑣 = 𝑦 → (∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑦) = (𝑎 · (𝑊‘𝐼))))
98		oveq1 7394	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (𝑎 · (𝑊‘𝐼)) = (𝑏 · (𝑊‘𝐼)))
99	98	eqeq2d 2740	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (((𝑃‘𝐼)‘𝑦) = (𝑎 · (𝑊‘𝐼)) ↔ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼))))
100	99	cbvrexvw 3216	. . . . . 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 3589	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ∃𝑏 ∈ ℤ ((𝑃‘𝐼)‘𝑦) = (𝑏 · (𝑊‘𝐼)))
104		reeanv 3209	. . . . 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 14071	. . . . . . . . 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 2729	. . . . . . . . . . 11 ⊢ (._r‘𝑍) = (._r‘𝑍)
116	4, 115	unitmulcl 20289	. . . . . . . . . 10 ⊢ ((𝑍 ∈ Ring ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈) → (𝑥(._r‘𝑍)𝑦) ∈ 𝑈)
117	112, 113, 114, 116	syl3anc 1373	. . . . . . . . 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 12642	. . . . . . . . 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 7405	. . . . . . . . . 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 19995	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃‘𝐼) ∈ ((𝐻 ↾_s (𝐻 DProd 𝑆)) GrpHom 𝐻))
123	21	oveq2d 7403	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐻 ↾_s (𝐻 DProd 𝑆)) = (𝐻 ↾_s 𝑈))
124	42	ovexi 7421	. . . . . . . . . . . . . . . 16 ⊢ 𝐻 ∈ V
125	69	ressid 17214	. . . . . . . . . . . . . . . 16 ⊢ (𝐻 ∈ V → (𝐻 ↾_s 𝑈) = 𝐻)
126	124, 125	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (𝐻 ↾_s 𝑈) = 𝐻
127	123, 126	eqtrdi 2780	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐻 ↾_s (𝐻 DProd 𝑆)) = 𝐻)
128	127	oveq1d 7402	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐻 ↾_s (𝐻 DProd 𝑆)) GrpHom 𝐻) = (𝐻 GrpHom 𝐻))
129	122, 128	eleqtrd 2830	. . . . . . . . . . . 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 6872	. . . . . . . . . . . . 13 ⊢ 𝑈 ∈ V
132		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
133	132, 115	mgpplusg 20053	. . . . . . . . . . . . . 14 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
134	42, 133	ressplusg 17254	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ V → (._r‘𝑍) = (+_g‘𝐻))
135	131, 134	ax-mp 5	. . . . . . . . . . . 12 ⊢ (._r‘𝑍) = (+_g‘𝐻)
136	69, 135, 135	ghmlin 19153	. . . . . . . . . . 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 1373	. . . . . . . . . 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 19038	. . . . . . . . . . 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 1374	. . . . . . . . . 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 2774	. . . . . . . . 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 27175	. . . . . . . . 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 27175	. . . . . . . . . 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 27175	. . . . . . . . . 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 7405	. . . . . . . 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 2774	. . . . . . 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 3194	. . . . 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 2729	. . . . . . 7 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
157	4, 156	1unit 20283	. . . . . 6 ⊢ (𝑍 ∈ Ring → (1_r‘𝑍) ∈ 𝑈)
158	56, 157	syl 17	. . . . 5 ⊢ (𝜑 → (1_r‘𝑍) ∈ 𝑈)
159		0zd 12541	. . . . 5 ⊢ (𝜑 → 0 ∈ ℤ)
160		eqid 2729	. . . . . . . 8 ⊢ (0_g‘𝐻) = (0_g‘𝐻)
161	160, 160	ghmid 19154	. . . . . . 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 20294	. . . . . . . 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 6862	. . . . . 6 ⊢ (𝜑 → ((𝑃‘𝐼)‘(1_r‘𝑍)) = ((𝑃‘𝐼)‘(0_g‘𝐻)))
166	69, 160, 43	mulg0 19006	. . . . . . 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 2774	. . . . 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 27175	. . . . 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 14105	. . . 4 ⊢ (𝜑 → (𝑇↑0) = 1)
172	170, 171	eqtrd 2764	. . 3 ⊢ (𝜑 → (𝑋‘(1_r‘𝑍)) = 1)
173	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 89, 154, 172	dchrelbasd 27150	. 2 ⊢ (𝜑 → (𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0)) ∈ 𝐷)
174	61, 44	sselid 3944	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
175		eleq1 2816	. . . . . . 7 ⊢ (𝑣 = 𝐴 → (𝑣 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈))
176		fveq2 6858	. . . . . . 7 ⊢ (𝑣 = 𝐴 → (𝑋‘𝑣) = (𝑋‘𝐴))
177	175, 176	ifbieq1d 4513	. . . . . 6 ⊢ (𝑣 = 𝐴 → if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0) = if(𝐴 ∈ 𝑈, (𝑋‘𝐴), 0))
178		eqid 2729	. . . . . 6 ⊢ (𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0)) = (𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0))
179		fvex 6871	. . . . . . 7 ⊢ (𝑋‘𝑣) ∈ V
180		c0ex 11168	. . . . . . 7 ⊢ 0 ∈ V
181	179, 180	ifex 4539	. . . . . 6 ⊢ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0) ∈ V
182	177, 178, 181	fvmpt3i 6973	. . . . 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 4496	. . . 4 ⊢ (𝜑 → if(𝐴 ∈ 𝑈, (𝑋‘𝐴), 0) = (𝑋‘𝐴))
185	183, 184	eqtrd 2764	. . 3 ⊢ (𝜑 → ((𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0))‘𝐴) = (𝑋‘𝐴))
186		fveqeq2 6867	. . . . . 6 ⊢ (𝑣 = 𝐴 → (((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼))))
187	186	rexbidv 3157	. . . . 5 ⊢ (𝑣 = 𝐴 → (∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝑣) = (𝑎 · (𝑊‘𝐼)) ↔ ∃𝑎 ∈ ℤ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼))))
188	187, 92, 44	rspcdva 3589	. . . 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 27175	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → (𝑋‘𝐴) = (𝑇↑𝑎))
190	47	oveq1i 7397	. . . . . . . 8 ⊢ (𝑇↑𝑎) = ((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑𝑎)
191	189, 190	eqtrdi 2780	. . . . . . 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 19477	. . . . . . . . . . 11 ⊢ ((𝐻 ∈ Grp ∧ (𝑊‘𝐼) ∈ 𝑈 ∧ 𝑎 ∈ ℤ) → ((𝑂‘(𝑊‘𝐼)) ∥ 𝑎 ↔ (𝑎 · (𝑊‘𝐼)) = (0_g‘𝐻)))
197	193, 194, 195, 196	syl3anc 1373	. . . . . . . . . 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 26665	. . . . . . . . . . 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 2776	. . . . . . . . . . . 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 2745	. . . . . . . . . 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 2969	. . . . . . . 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 2992	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑈) ∧ (𝑎 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐴) = (𝑎 · (𝑊‘𝐼)))) → (𝑋‘𝐴) ≠ 1)
209	208	rexlimdvaa 3135	. . . . 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 2992	. 2 ⊢ (𝜑 → ((𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0))‘𝐴) ≠ 1)
213		fveq1 6857	. . . 4 ⊢ (𝑥 = (𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0)) → (𝑥‘𝐴) = ((𝑣 ∈ 𝐵 ↦ if(𝑣 ∈ 𝑈, (𝑋‘𝑣), 0))‘𝐴))
214	213	neeq1d 2984	. . 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 3588	. 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 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 Vcvv 3447 ⊆ wss 3914 ifcif 4488 class class class wbr 5107 ↦ cmpt 5188 dom cdm 5638 ran crn 5639 ℩cio 6462 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 Fincfn 8918 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 -cneg 11406 / cdiv 11835 ℕcn 12186 2c2 12241 ℕ₀cn0 12442 ℤcz 12529 ..^cfzo 13615 ↑cexp 14026 ♯chash 14295 Word cword 14478 ∥ cdvds 16222 Basecbs 17179 ↾_s cress 17200 +_gcplusg 17220 ._rcmulr 17221 0_gc0g 17402 Grpcgrp 18865 ._gcmg 18999 GrpHom cghm 19144 odcod 19454 DProd cdprd 19925 dProjcdpj 19926 mulGrpcmgp 20049 1_rcur 20090 Ringcrg 20142 CRingccrg 20143 Unitcui 20264 ℤ/nℤczn 21412 ↑_𝑐ccxp 26464 DChrcdchr 27143

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147 ax-mulf 11148

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-omul 8439 df-er 8671 df-ec 8673 df-qs 8677 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-acn 9895 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-word 14479 df-shft 15033 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-sum 15653 df-ef 16033 df-sin 16035 df-cos 16036 df-pi 16038 df-dvds 16223 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-qus 17472 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-nsg 19056 df-eqg 19057 df-ghm 19145 df-gim 19191 df-cntz 19249 df-oppg 19278 df-od 19458 df-lsm 19566 df-pj1 19567 df-cmn 19712 df-abl 19713 df-dprd 19927 df-dpj 19928 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-cring 20145 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-rhm 20381 df-subrng 20455 df-subrg 20479 df-lmod 20768 df-lss 20838 df-lsp 20878 df-sra 21080 df-rgmod 21081 df-lidl 21118 df-rsp 21119 df-2idl 21160 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-zring 21357 df-zrh 21413 df-zn 21416 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-haus 23202 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-limc 25767 df-dv 25768 df-log 26465 df-cxp 26466 df-dchr 27144

This theorem is referenced by: dchrptlem3 27177

W3C validator