prjspner1 - Mathbox for Steven Nguyen

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Theorem prjspner1 41946

Description: Two vectors whose zeroth coordinate is nonzero are equivalent if and only if they have the same representative in the (n-1)-dimensional affine subspace { x₀ = 1 } . For example, vectors in 3D space whose 𝑥 coordinate is nonzero are equivalent iff they intersect at the plane 𝑥 = 1 at the same point (also see section header). (Contributed by SN, 13-Aug-2023.)

**Hypotheses**
Ref	Expression
prjspner01.e	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}
prjspner01.f	⊢ 𝐹 = (𝑏 ∈ 𝐵 ↦ if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)))
prjspner01.b	⊢ 𝐵 = ((Base‘𝑊) ∖ {(0_g‘𝑊)})
prjspner01.w	⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))
prjspner01.t	⊢ · = ( ·_𝑠 ‘𝑊)
prjspner01.s	⊢ 𝑆 = (Base‘𝐾)
prjspner01.0	⊢ 0 = (0_g‘𝐾)
prjspner01.i	⊢ 𝐼 = (inv_r‘𝐾)
prjspner01.k	⊢ (𝜑 → 𝐾 ∈ DivRing)
prjspner01.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
prjspner01.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
prjspner1.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
prjspner1.1	⊢ (𝜑 → (𝑋‘0) ≠ 0 )
prjspner1.2	⊢ (𝜑 → (𝑌‘0) ≠ 0 )

**Assertion**
Ref	Expression
prjspner1	⊢ (𝜑 → (𝑋 ∼ 𝑌 ↔ (𝐹‘𝑋) = (𝐹‘𝑌)))

Distinct variable groups: 𝑥,𝐵,𝑦 𝑋,𝑙,𝑥,𝑦 𝑊,𝑙,𝑥,𝑦 · ,𝑙,𝑥,𝑦 𝑆,𝑙 𝐼,𝑙,𝑥,𝑦 𝑥,𝐾,𝑦 𝑥, 0 ,𝑦 𝐵,𝑏 𝑋,𝑏 0 ,𝑏 · ,𝑏 𝐼,𝑏 𝜑,𝑏 𝑌,𝑙,𝑥,𝑦 𝑌,𝑏 𝑥,𝑆,𝑦 Allowed substitution hints: 𝜑(𝑥,𝑦,𝑙) 𝐵(𝑙) ∼ (𝑥,𝑦,𝑏,𝑙) 𝑆(𝑏) 𝐹(𝑥,𝑦,𝑏,𝑙) 𝐾(𝑏,𝑙) 𝑁(𝑥,𝑦,𝑏,𝑙) 𝑊(𝑏) 0 (𝑙)

Proof of Theorem prjspner1 Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prjspner01.e	. . . 4 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}
2	1	prjsprel 41924	. . 3 ⊢ (𝑋 ∼ 𝑌 ↔ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∃𝑚 ∈ 𝑆 𝑋 = (𝑚 · 𝑌)))
3		prjspner1.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋‘0) ≠ 0 )
4		fveq1 6884	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = (0_g‘𝑊) → (𝑋‘0) = ((0_g‘𝑊)‘0))
5		prjspner01.w	. . . . . . . . . . . . . . . 16 ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))
6		prjspner01.0	. . . . . . . . . . . . . . . 16 ⊢ 0 = (0_g‘𝐾)
7		prjspner01.k	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 ∈ DivRing)
8	7	drngringd 20595	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ Ring)
9		ovexd 7440	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0...𝑁) ∈ V)
10		prjspner01.n	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
11		0elfz 13604	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ₀ → 0 ∈ (0...𝑁))
12	10, 11	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ∈ (0...𝑁))
13	5, 6, 8, 9, 12	frlm0vald 41666	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((0_g‘𝑊)‘0) = 0 )
14	4, 13	sylan9eqr 2788	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑊)) → (𝑋‘0) = 0 )
15	3, 14	mteqand 3027	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ≠ (0_g‘𝑊))
16	5	frlmsca 21648	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ DivRing ∧ (0...𝑁) ∈ V) → 𝐾 = (Scalar‘𝑊))
17	7, 9, 16	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 = (Scalar‘𝑊))
18	17	fveq2d 6889	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0_g‘𝐾) = (0_g‘(Scalar‘𝑊)))
19	6, 18	eqtrid 2778	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 = (0_g‘(Scalar‘𝑊)))
20	19	oveq1d 7420	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ( 0 · 𝑌) = ((0_g‘(Scalar‘𝑊)) · 𝑌))
21	5	frlmlvec 21656	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ DivRing ∧ (0...𝑁) ∈ V) → 𝑊 ∈ LVec)
22	7, 9, 21	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑊 ∈ LVec)
23	22	lveclmodd 20955	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑊 ∈ LMod)
24		prjspner1.y	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
25		prjspner01.b	. . . . . . . . . . . . . . . . 17 ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0_g‘𝑊)})
26	24, 25	eleqtrdi 2837	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ∈ ((Base‘𝑊) ∖ {(0_g‘𝑊)}))
27	26	eldifad 3955	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑊))
28		eqid 2726	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑊) = (Base‘𝑊)
29		eqid 2726	. . . . . . . . . . . . . . . 16 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
30		prjspner01.t	. . . . . . . . . . . . . . . 16 ⊢ · = ( ·_𝑠 ‘𝑊)
31		eqid 2726	. . . . . . . . . . . . . . . 16 ⊢ (0_g‘(Scalar‘𝑊)) = (0_g‘(Scalar‘𝑊))
32		eqid 2726	. . . . . . . . . . . . . . . 16 ⊢ (0_g‘𝑊) = (0_g‘𝑊)
33	28, 29, 30, 31, 32	lmod0vs 20741	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ (Base‘𝑊)) → ((0_g‘(Scalar‘𝑊)) · 𝑌) = (0_g‘𝑊))
34	23, 27, 33	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((0_g‘(Scalar‘𝑊)) · 𝑌) = (0_g‘𝑊))
35	20, 34	eqtrd 2766	. . . . . . . . . . . . 13 ⊢ (𝜑 → ( 0 · 𝑌) = (0_g‘𝑊))
36	15, 35	neeqtrrd 3009	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ≠ ( 0 · 𝑌))
37	36	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) → 𝑋 ≠ ( 0 · 𝑌))
38		oveq1 7412	. . . . . . . . . . . 12 ⊢ (𝑚 = 0 → (𝑚 · 𝑌) = ( 0 · 𝑌))
39	38	neeq2d 2995	. . . . . . . . . . 11 ⊢ (𝑚 = 0 → (𝑋 ≠ (𝑚 · 𝑌) ↔ 𝑋 ≠ ( 0 · 𝑌)))
40	37, 39	syl5ibrcom 246	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) → (𝑚 = 0 → 𝑋 ≠ (𝑚 · 𝑌)))
41	40	necon2d 2957	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) → (𝑋 = (𝑚 · 𝑌) → 𝑚 ≠ 0 ))
42	41	ancrd 551	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) → (𝑋 = (𝑚 · 𝑌) → (𝑚 ≠ 0 ∧ 𝑋 = (𝑚 · 𝑌))))
43		prjspner01.s	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (Base‘𝐾)
44		ovexd 7440	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → (0...𝑁) ∈ V)
45		simplr 766	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → 𝑚 ∈ 𝑆)
46	27	ad3antrrr 727	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → 𝑌 ∈ (Base‘𝑊))
47	12	ad3antrrr 727	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → 0 ∈ (0...𝑁))
48		eqid 2726	. . . . . . . . . . . . . . 15 ⊢ (._r‘𝐾) = (._r‘𝐾)
49	5, 28, 43, 44, 45, 46, 47, 30, 48	frlmvscaval 21663	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → ((𝑚 · 𝑌)‘0) = (𝑚(._r‘𝐾)(𝑌‘0)))
50	49	fveq2d 6889	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → (𝐼‘((𝑚 · 𝑌)‘0)) = (𝐼‘(𝑚(._r‘𝐾)(𝑌‘0))))
51		prjspner01.i	. . . . . . . . . . . . . 14 ⊢ 𝐼 = (inv_r‘𝐾)
52	7	ad3antrrr 727	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → 𝐾 ∈ DivRing)
53	5, 43, 28	frlmbasf 21655	. . . . . . . . . . . . . . . . 17 ⊢ (((0...𝑁) ∈ V ∧ 𝑌 ∈ (Base‘𝑊)) → 𝑌:(0...𝑁)⟶𝑆)
54	9, 27, 53	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌:(0...𝑁)⟶𝑆)
55	54, 12	ffvelcdmd 7081	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑌‘0) ∈ 𝑆)
56	55	ad3antrrr 727	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → (𝑌‘0) ∈ 𝑆)
57		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → 𝑚 ≠ 0 )
58		prjspner1.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑌‘0) ≠ 0 )
59	58	ad3antrrr 727	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → (𝑌‘0) ≠ 0 )
60	43, 6, 48, 51, 52, 45, 56, 57, 59	drnginvmuld 41660	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → (𝐼‘(𝑚(._r‘𝐾)(𝑌‘0))) = ((𝐼‘(𝑌‘0))(._r‘𝐾)(𝐼‘𝑚)))
61	50, 60	eqtrd 2766	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → (𝐼‘((𝑚 · 𝑌)‘0)) = ((𝐼‘(𝑌‘0))(._r‘𝐾)(𝐼‘𝑚)))
62	61	oveq1d 7420	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → ((𝐼‘((𝑚 · 𝑌)‘0)) · (𝑚 · 𝑌)) = (((𝐼‘(𝑌‘0))(._r‘𝐾)(𝐼‘𝑚)) · (𝑚 · 𝑌)))
63	23	ad3antrrr 727	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → 𝑊 ∈ LMod)
64	8	ad3antrrr 727	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → 𝐾 ∈ Ring)
65	43, 6, 51, 52, 56, 59	drnginvrcld 20611	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → (𝐼‘(𝑌‘0)) ∈ 𝑆)
66	43, 6, 51, 52, 45, 57	drnginvrcld 20611	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → (𝐼‘𝑚) ∈ 𝑆)
67	43, 48, 64, 65, 66	ringcld 20162	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → ((𝐼‘(𝑌‘0))(._r‘𝐾)(𝐼‘𝑚)) ∈ 𝑆)
68	17	fveq2d 6889	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘𝑊)))
69	43, 68	eqtrid 2778	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 = (Base‘(Scalar‘𝑊)))
70	69	ad3antrrr 727	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → 𝑆 = (Base‘(Scalar‘𝑊)))
71	67, 70	eleqtrd 2829	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → ((𝐼‘(𝑌‘0))(._r‘𝐾)(𝐼‘𝑚)) ∈ (Base‘(Scalar‘𝑊)))
72	45, 70	eleqtrd 2829	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → 𝑚 ∈ (Base‘(Scalar‘𝑊)))
73		eqid 2726	. . . . . . . . . . . . 13 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
74		eqid 2726	. . . . . . . . . . . . 13 ⊢ (._r‘(Scalar‘𝑊)) = (._r‘(Scalar‘𝑊))
75	28, 29, 30, 73, 74	lmodvsass 20733	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ (((𝐼‘(𝑌‘0))(._r‘𝐾)(𝐼‘𝑚)) ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑚 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ (Base‘𝑊))) → ((((𝐼‘(𝑌‘0))(._r‘𝐾)(𝐼‘𝑚))(._r‘(Scalar‘𝑊))𝑚) · 𝑌) = (((𝐼‘(𝑌‘0))(._r‘𝐾)(𝐼‘𝑚)) · (𝑚 · 𝑌)))
76	63, 71, 72, 46, 75	syl13anc 1369	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → ((((𝐼‘(𝑌‘0))(._r‘𝐾)(𝐼‘𝑚))(._r‘(Scalar‘𝑊))𝑚) · 𝑌) = (((𝐼‘(𝑌‘0))(._r‘𝐾)(𝐼‘𝑚)) · (𝑚 · 𝑌)))
77	43, 48, 64, 65, 66, 45	ringassd 20161	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → (((𝐼‘(𝑌‘0))(._r‘𝐾)(𝐼‘𝑚))(._r‘𝐾)𝑚) = ((𝐼‘(𝑌‘0))(._r‘𝐾)((𝐼‘𝑚)(._r‘𝐾)𝑚)))
78	52, 44, 16	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → 𝐾 = (Scalar‘𝑊))
79	78	fveq2d 6889	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → (._r‘𝐾) = (._r‘(Scalar‘𝑊)))
80	79	oveqd 7422	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → (((𝐼‘(𝑌‘0))(._r‘𝐾)(𝐼‘𝑚))(._r‘𝐾)𝑚) = (((𝐼‘(𝑌‘0))(._r‘𝐾)(𝐼‘𝑚))(._r‘(Scalar‘𝑊))𝑚))
81		eqid 2726	. . . . . . . . . . . . . . . 16 ⊢ (1_r‘𝐾) = (1_r‘𝐾)
82	43, 6, 48, 81, 51, 52, 45, 57	drnginvrld 20614	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → ((𝐼‘𝑚)(._r‘𝐾)𝑚) = (1_r‘𝐾))
83	82	oveq2d 7421	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → ((𝐼‘(𝑌‘0))(._r‘𝐾)((𝐼‘𝑚)(._r‘𝐾)𝑚)) = ((𝐼‘(𝑌‘0))(._r‘𝐾)(1_r‘𝐾)))
84	43, 48, 81, 64, 65	ringridmd 20172	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → ((𝐼‘(𝑌‘0))(._r‘𝐾)(1_r‘𝐾)) = (𝐼‘(𝑌‘0)))
85	83, 84	eqtrd 2766	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → ((𝐼‘(𝑌‘0))(._r‘𝐾)((𝐼‘𝑚)(._r‘𝐾)𝑚)) = (𝐼‘(𝑌‘0)))
86	77, 80, 85	3eqtr3d 2774	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → (((𝐼‘(𝑌‘0))(._r‘𝐾)(𝐼‘𝑚))(._r‘(Scalar‘𝑊))𝑚) = (𝐼‘(𝑌‘0)))
87	86	oveq1d 7420	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → ((((𝐼‘(𝑌‘0))(._r‘𝐾)(𝐼‘𝑚))(._r‘(Scalar‘𝑊))𝑚) · 𝑌) = ((𝐼‘(𝑌‘0)) · 𝑌))
88	62, 76, 87	3eqtr2d 2772	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → ((𝐼‘((𝑚 · 𝑌)‘0)) · (𝑚 · 𝑌)) = ((𝐼‘(𝑌‘0)) · 𝑌))
89		fveq1 6884	. . . . . . . . . . . . 13 ⊢ (𝑋 = (𝑚 · 𝑌) → (𝑋‘0) = ((𝑚 · 𝑌)‘0))
90	89	fveq2d 6889	. . . . . . . . . . . 12 ⊢ (𝑋 = (𝑚 · 𝑌) → (𝐼‘(𝑋‘0)) = (𝐼‘((𝑚 · 𝑌)‘0)))
91		id 22	. . . . . . . . . . . 12 ⊢ (𝑋 = (𝑚 · 𝑌) → 𝑋 = (𝑚 · 𝑌))
92	90, 91	oveq12d 7423	. . . . . . . . . . 11 ⊢ (𝑋 = (𝑚 · 𝑌) → ((𝐼‘(𝑋‘0)) · 𝑋) = ((𝐼‘((𝑚 · 𝑌)‘0)) · (𝑚 · 𝑌)))
93	92	eqeq1d 2728	. . . . . . . . . 10 ⊢ (𝑋 = (𝑚 · 𝑌) → (((𝐼‘(𝑋‘0)) · 𝑋) = ((𝐼‘(𝑌‘0)) · 𝑌) ↔ ((𝐼‘((𝑚 · 𝑌)‘0)) · (𝑚 · 𝑌)) = ((𝐼‘(𝑌‘0)) · 𝑌)))
94	88, 93	syl5ibrcom 246	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) ∧ 𝑚 ≠ 0 ) → (𝑋 = (𝑚 · 𝑌) → ((𝐼‘(𝑋‘0)) · 𝑋) = ((𝐼‘(𝑌‘0)) · 𝑌)))
95	94	expimpd 453	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) → ((𝑚 ≠ 0 ∧ 𝑋 = (𝑚 · 𝑌)) → ((𝐼‘(𝑋‘0)) · 𝑋) = ((𝐼‘(𝑌‘0)) · 𝑌)))
96	42, 95	syld 47	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑚 ∈ 𝑆) → (𝑋 = (𝑚 · 𝑌) → ((𝐼‘(𝑋‘0)) · 𝑋) = ((𝐼‘(𝑌‘0)) · 𝑌)))
97	96	rexlimdva 3149	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (∃𝑚 ∈ 𝑆 𝑋 = (𝑚 · 𝑌) → ((𝐼‘(𝑋‘0)) · 𝑋) = ((𝐼‘(𝑌‘0)) · 𝑌)))
98	97	impr 454	. . . . 5 ⊢ ((𝜑 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∃𝑚 ∈ 𝑆 𝑋 = (𝑚 · 𝑌))) → ((𝐼‘(𝑋‘0)) · 𝑋) = ((𝐼‘(𝑌‘0)) · 𝑌))
99	3	neneqd 2939	. . . . . . 7 ⊢ (𝜑 → ¬ (𝑋‘0) = 0 )
100	99	iffalsed 4534	. . . . . 6 ⊢ (𝜑 → if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋)) = ((𝐼‘(𝑋‘0)) · 𝑋))
101	100	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∃𝑚 ∈ 𝑆 𝑋 = (𝑚 · 𝑌))) → if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋)) = ((𝐼‘(𝑋‘0)) · 𝑋))
102	58	neneqd 2939	. . . . . . 7 ⊢ (𝜑 → ¬ (𝑌‘0) = 0 )
103	102	iffalsed 4534	. . . . . 6 ⊢ (𝜑 → if((𝑌‘0) = 0 , 𝑌, ((𝐼‘(𝑌‘0)) · 𝑌)) = ((𝐼‘(𝑌‘0)) · 𝑌))
104	103	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∃𝑚 ∈ 𝑆 𝑋 = (𝑚 · 𝑌))) → if((𝑌‘0) = 0 , 𝑌, ((𝐼‘(𝑌‘0)) · 𝑌)) = ((𝐼‘(𝑌‘0)) · 𝑌))
105	98, 101, 104	3eqtr4d 2776	. . . 4 ⊢ ((𝜑 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∃𝑚 ∈ 𝑆 𝑋 = (𝑚 · 𝑌))) → if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋)) = if((𝑌‘0) = 0 , 𝑌, ((𝐼‘(𝑌‘0)) · 𝑌)))
106		prjspner01.f	. . . . 5 ⊢ 𝐹 = (𝑏 ∈ 𝐵 ↦ if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)))
107		fveq1 6884	. . . . . . 7 ⊢ (𝑏 = 𝑋 → (𝑏‘0) = (𝑋‘0))
108	107	eqeq1d 2728	. . . . . 6 ⊢ (𝑏 = 𝑋 → ((𝑏‘0) = 0 ↔ (𝑋‘0) = 0 ))
109		id 22	. . . . . 6 ⊢ (𝑏 = 𝑋 → 𝑏 = 𝑋)
110	107	fveq2d 6889	. . . . . . 7 ⊢ (𝑏 = 𝑋 → (𝐼‘(𝑏‘0)) = (𝐼‘(𝑋‘0)))
111	110, 109	oveq12d 7423	. . . . . 6 ⊢ (𝑏 = 𝑋 → ((𝐼‘(𝑏‘0)) · 𝑏) = ((𝐼‘(𝑋‘0)) · 𝑋))
112	108, 109, 111	ifbieq12d 4551	. . . . 5 ⊢ (𝑏 = 𝑋 → if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)) = if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋)))
113		simprll 776	. . . . 5 ⊢ ((𝜑 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∃𝑚 ∈ 𝑆 𝑋 = (𝑚 · 𝑌))) → 𝑋 ∈ 𝐵)
114		ovexd 7440	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∃𝑚 ∈ 𝑆 𝑋 = (𝑚 · 𝑌))) → ((𝐼‘(𝑋‘0)) · 𝑋) ∈ V)
115	113, 114	ifexd 4571	. . . . 5 ⊢ ((𝜑 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∃𝑚 ∈ 𝑆 𝑋 = (𝑚 · 𝑌))) → if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋)) ∈ V)
116	106, 112, 113, 115	fvmptd3 7015	. . . 4 ⊢ ((𝜑 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∃𝑚 ∈ 𝑆 𝑋 = (𝑚 · 𝑌))) → (𝐹‘𝑋) = if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋)))
117		fveq1 6884	. . . . . . 7 ⊢ (𝑏 = 𝑌 → (𝑏‘0) = (𝑌‘0))
118	117	eqeq1d 2728	. . . . . 6 ⊢ (𝑏 = 𝑌 → ((𝑏‘0) = 0 ↔ (𝑌‘0) = 0 ))
119		id 22	. . . . . 6 ⊢ (𝑏 = 𝑌 → 𝑏 = 𝑌)
120	117	fveq2d 6889	. . . . . . 7 ⊢ (𝑏 = 𝑌 → (𝐼‘(𝑏‘0)) = (𝐼‘(𝑌‘0)))
121	120, 119	oveq12d 7423	. . . . . 6 ⊢ (𝑏 = 𝑌 → ((𝐼‘(𝑏‘0)) · 𝑏) = ((𝐼‘(𝑌‘0)) · 𝑌))
122	118, 119, 121	ifbieq12d 4551	. . . . 5 ⊢ (𝑏 = 𝑌 → if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)) = if((𝑌‘0) = 0 , 𝑌, ((𝐼‘(𝑌‘0)) · 𝑌)))
123		simprlr 777	. . . . 5 ⊢ ((𝜑 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∃𝑚 ∈ 𝑆 𝑋 = (𝑚 · 𝑌))) → 𝑌 ∈ 𝐵)
124		ovexd 7440	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∃𝑚 ∈ 𝑆 𝑋 = (𝑚 · 𝑌))) → ((𝐼‘(𝑌‘0)) · 𝑌) ∈ V)
125	123, 124	ifexd 4571	. . . . 5 ⊢ ((𝜑 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∃𝑚 ∈ 𝑆 𝑋 = (𝑚 · 𝑌))) → if((𝑌‘0) = 0 , 𝑌, ((𝐼‘(𝑌‘0)) · 𝑌)) ∈ V)
126	106, 122, 123, 125	fvmptd3 7015	. . . 4 ⊢ ((𝜑 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∃𝑚 ∈ 𝑆 𝑋 = (𝑚 · 𝑌))) → (𝐹‘𝑌) = if((𝑌‘0) = 0 , 𝑌, ((𝐼‘(𝑌‘0)) · 𝑌)))
127	105, 116, 126	3eqtr4d 2776	. . 3 ⊢ ((𝜑 ∧ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∃𝑚 ∈ 𝑆 𝑋 = (𝑚 · 𝑌))) → (𝐹‘𝑋) = (𝐹‘𝑌))
128	2, 127	sylan2b 593	. 2 ⊢ ((𝜑 ∧ 𝑋 ∼ 𝑌) → (𝐹‘𝑋) = (𝐹‘𝑌))
129	1, 5, 25, 43, 30, 7	prjspner 41939	. . . 4 ⊢ (𝜑 → ∼ Er 𝐵)
130	129	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑋) = (𝐹‘𝑌)) → ∼ Er 𝐵)
131		prjspner01.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
132	1, 106, 25, 5, 30, 43, 6, 51, 7, 10, 131	prjspner01 41945	. . . 4 ⊢ (𝜑 → 𝑋 ∼ (𝐹‘𝑋))
133	132	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑋) = (𝐹‘𝑌)) → 𝑋 ∼ (𝐹‘𝑋))
134	129, 132	ercl2 8718	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐵)
135	134	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹‘𝑋) = (𝐹‘𝑌)) → (𝐹‘𝑋) ∈ 𝐵)
136	130, 135	erref 8725	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝑋) = (𝐹‘𝑌)) → (𝐹‘𝑋) ∼ (𝐹‘𝑋))
137		breq2 5145	. . . . . 6 ⊢ ((𝐹‘𝑋) = (𝐹‘𝑌) → ((𝐹‘𝑋) ∼ (𝐹‘𝑋) ↔ (𝐹‘𝑋) ∼ (𝐹‘𝑌)))
138	137	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝑋) = (𝐹‘𝑌)) → ((𝐹‘𝑋) ∼ (𝐹‘𝑋) ↔ (𝐹‘𝑋) ∼ (𝐹‘𝑌)))
139	136, 138	mpbid 231	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝑋) = (𝐹‘𝑌)) → (𝐹‘𝑋) ∼ (𝐹‘𝑌))
140	1, 106, 25, 5, 30, 43, 6, 51, 7, 10, 24	prjspner01 41945	. . . . 5 ⊢ (𝜑 → 𝑌 ∼ (𝐹‘𝑌))
141	140	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝑋) = (𝐹‘𝑌)) → 𝑌 ∼ (𝐹‘𝑌))
142	130, 139, 141	ertr4d 8724	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑋) = (𝐹‘𝑌)) → (𝐹‘𝑋) ∼ 𝑌)
143	130, 133, 142	ertrd 8721	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝑋) = (𝐹‘𝑌)) → 𝑋 ∼ 𝑌)
144	128, 143	impbida 798	1 ⊢ (𝜑 → (𝑋 ∼ 𝑌 ↔ (𝐹‘𝑋) = (𝐹‘𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∃wrex 3064 Vcvv 3468 ∖ cdif 3940 ifcif 4523 {csn 4623 class class class wbr 5141 {copab 5203 ↦ cmpt 5224 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 Er wer 8702 0cc0 11112 ℕ₀cn0 12476 ...cfz 13490 Basecbs 17153 ._rcmulr 17207 Scalarcsca 17209 ·_𝑠 cvsca 17210 0_gc0g 17394 1_rcur 20086 Ringcrg 20138 inv_rcinvr 20289 DivRingcdr 20587 LModclmod 20706 LVecclvec 20950 freeLMod cfrlm 21641

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ifp 1060 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-hom 17230 df-cco 17231 df-0g 17396 df-prds 17402 df-pws 17404 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19050 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-subrg 20471 df-drng 20589 df-lmod 20708 df-lss 20779 df-lvec 20951 df-sra 21021 df-rgmod 21022 df-dsmm 21627 df-frlm 21642

This theorem is referenced by: (None)

W3C validator