line2y - Mathbox for Alexander van der Vekens

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Theorem line2y 44762

Description: Example for a vertical line 𝐺 passing through two different points in "standard form". (Contributed by AV, 3-Feb-2023.)

**Hypotheses**
Ref	Expression
line2.i	⊢ 𝐼 = {1, 2}
line2.e	⊢ 𝐸 = (ℝ^‘𝐼)
line2.p	⊢ 𝑃 = (ℝ ↑_m 𝐼)
line2.l	⊢ 𝐿 = (Line_M‘𝐸)
line2.g	⊢ 𝐺 = {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}
line2y.x	⊢ 𝑋 = {⟨1, 0⟩, ⟨2, 𝑀⟩}
line2y.y	⊢ 𝑌 = {⟨1, 0⟩, ⟨2, 𝑁⟩}

**Assertion**
Ref	Expression
line2y	⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) → (𝐺 = (𝑋𝐿𝑌) ↔ (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)))

Distinct variable groups: 𝐴,𝑝 𝐵,𝑝 𝐶,𝑝 𝐸,𝑝 𝐼,𝑝 𝑃,𝑝 𝑋,𝑝 𝑌,𝑝 𝑀,𝑝 𝑁,𝑝 Allowed substitution hints: 𝐺(𝑝) 𝐿(𝑝)

**Proof of Theorem line2y**
Step	Hyp	Ref	Expression
1		line2.g	. . . 4 ⊢ 𝐺 = {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}
2	1	a1i 11	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) → 𝐺 = {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶})
3		1ex 10637	. . . . . . . . . . . 12 ⊢ 1 ∈ V
4		2ex 11715	. . . . . . . . . . . 12 ⊢ 2 ∈ V
5	3, 4	pm3.2i 473	. . . . . . . . . . 11 ⊢ (1 ∈ V ∧ 2 ∈ V)
6		c0ex 10635	. . . . . . . . . . . 12 ⊢ 0 ∈ V
7	6	jctl 526	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℝ → (0 ∈ V ∧ 𝑀 ∈ ℝ))
8		1ne2 11846	. . . . . . . . . . . 12 ⊢ 1 ≠ 2
9	8	a1i 11	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℝ → 1 ≠ 2)
10		fprg 6917	. . . . . . . . . . . 12 ⊢ (((1 ∈ V ∧ 2 ∈ V) ∧ (0 ∈ V ∧ 𝑀 ∈ ℝ) ∧ 1 ≠ 2) → {⟨1, 0⟩, ⟨2, 𝑀⟩}:{1, 2}⟶{0, 𝑀})
11		0red 10644	. . . . . . . . . . . . 13 ⊢ (((1 ∈ V ∧ 2 ∈ V) ∧ (0 ∈ V ∧ 𝑀 ∈ ℝ) ∧ 1 ≠ 2) → 0 ∈ ℝ)
12		simp2r 1196	. . . . . . . . . . . . 13 ⊢ (((1 ∈ V ∧ 2 ∈ V) ∧ (0 ∈ V ∧ 𝑀 ∈ ℝ) ∧ 1 ≠ 2) → 𝑀 ∈ ℝ)
13	11, 12	prssd 4755	. . . . . . . . . . . 12 ⊢ (((1 ∈ V ∧ 2 ∈ V) ∧ (0 ∈ V ∧ 𝑀 ∈ ℝ) ∧ 1 ≠ 2) → {0, 𝑀} ⊆ ℝ)
14	10, 13	fssd 6528	. . . . . . . . . . 11 ⊢ (((1 ∈ V ∧ 2 ∈ V) ∧ (0 ∈ V ∧ 𝑀 ∈ ℝ) ∧ 1 ≠ 2) → {⟨1, 0⟩, ⟨2, 𝑀⟩}:{1, 2}⟶ℝ)
15	5, 7, 9, 14	mp3an2i 1462	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℝ → {⟨1, 0⟩, ⟨2, 𝑀⟩}:{1, 2}⟶ℝ)
16		line2.i	. . . . . . . . . . 11 ⊢ 𝐼 = {1, 2}
17	16	feq2i 6506	. . . . . . . . . 10 ⊢ ({⟨1, 0⟩, ⟨2, 𝑀⟩}:𝐼⟶ℝ ↔ {⟨1, 0⟩, ⟨2, 𝑀⟩}:{1, 2}⟶ℝ)
18	15, 17	sylibr 236	. . . . . . . . 9 ⊢ (𝑀 ∈ ℝ → {⟨1, 0⟩, ⟨2, 𝑀⟩}:𝐼⟶ℝ)
19		reex 10628	. . . . . . . . . 10 ⊢ ℝ ∈ V
20		prex 5333	. . . . . . . . . . 11 ⊢ {1, 2} ∈ V
21	16, 20	eqeltri 2909	. . . . . . . . . 10 ⊢ 𝐼 ∈ V
22	19, 21	elmap 8435	. . . . . . . . 9 ⊢ ({⟨1, 0⟩, ⟨2, 𝑀⟩} ∈ (ℝ ↑_m 𝐼) ↔ {⟨1, 0⟩, ⟨2, 𝑀⟩}:𝐼⟶ℝ)
23	18, 22	sylibr 236	. . . . . . . 8 ⊢ (𝑀 ∈ ℝ → {⟨1, 0⟩, ⟨2, 𝑀⟩} ∈ (ℝ ↑_m 𝐼))
24		line2y.x	. . . . . . . 8 ⊢ 𝑋 = {⟨1, 0⟩, ⟨2, 𝑀⟩}
25		line2.p	. . . . . . . 8 ⊢ 𝑃 = (ℝ ↑_m 𝐼)
26	23, 24, 25	3eltr4g 2930	. . . . . . 7 ⊢ (𝑀 ∈ ℝ → 𝑋 ∈ 𝑃)
27	26	3ad2ant1 1129	. . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → 𝑋 ∈ 𝑃)
28	6	jctl 526	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → (0 ∈ V ∧ 𝑁 ∈ ℝ))
29	8	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → 1 ≠ 2)
30		fprg 6917	. . . . . . . . . . . 12 ⊢ (((1 ∈ V ∧ 2 ∈ V) ∧ (0 ∈ V ∧ 𝑁 ∈ ℝ) ∧ 1 ≠ 2) → {⟨1, 0⟩, ⟨2, 𝑁⟩}:{1, 2}⟶{0, 𝑁})
31	5, 28, 29, 30	mp3an2i 1462	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℝ → {⟨1, 0⟩, ⟨2, 𝑁⟩}:{1, 2}⟶{0, 𝑁})
32		0re 10643	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
33		prssi 4754	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ) → {0, 𝑁} ⊆ ℝ)
34	32, 33	mpan 688	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℝ → {0, 𝑁} ⊆ ℝ)
35	31, 34	fssd 6528	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ → {⟨1, 0⟩, ⟨2, 𝑁⟩}:{1, 2}⟶ℝ)
36	16	feq2i 6506	. . . . . . . . . 10 ⊢ ({⟨1, 0⟩, ⟨2, 𝑁⟩}:𝐼⟶ℝ ↔ {⟨1, 0⟩, ⟨2, 𝑁⟩}:{1, 2}⟶ℝ)
37	35, 36	sylibr 236	. . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → {⟨1, 0⟩, ⟨2, 𝑁⟩}:𝐼⟶ℝ)
38	19, 21	pm3.2i 473	. . . . . . . . . 10 ⊢ (ℝ ∈ V ∧ 𝐼 ∈ V)
39		elmapg 8419	. . . . . . . . . 10 ⊢ ((ℝ ∈ V ∧ 𝐼 ∈ V) → ({⟨1, 0⟩, ⟨2, 𝑁⟩} ∈ (ℝ ↑_m 𝐼) ↔ {⟨1, 0⟩, ⟨2, 𝑁⟩}:𝐼⟶ℝ))
40	38, 39	mp1i 13	. . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → ({⟨1, 0⟩, ⟨2, 𝑁⟩} ∈ (ℝ ↑_m 𝐼) ↔ {⟨1, 0⟩, ⟨2, 𝑁⟩}:𝐼⟶ℝ))
41	37, 40	mpbird 259	. . . . . . . 8 ⊢ (𝑁 ∈ ℝ → {⟨1, 0⟩, ⟨2, 𝑁⟩} ∈ (ℝ ↑_m 𝐼))
42		line2y.y	. . . . . . . 8 ⊢ 𝑌 = {⟨1, 0⟩, ⟨2, 𝑁⟩}
43	41, 42, 25	3eltr4g 2930	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → 𝑌 ∈ 𝑃)
44	43	3ad2ant2 1130	. . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → 𝑌 ∈ 𝑃)
45	24	fveq1i 6671	. . . . . . . . 9 ⊢ (𝑋‘1) = ({⟨1, 0⟩, ⟨2, 𝑀⟩}‘1)
46	3, 6, 8	3pm3.2i 1335	. . . . . . . . . 10 ⊢ (1 ∈ V ∧ 0 ∈ V ∧ 1 ≠ 2)
47		fvpr1g 6954	. . . . . . . . . 10 ⊢ ((1 ∈ V ∧ 0 ∈ V ∧ 1 ≠ 2) → ({⟨1, 0⟩, ⟨2, 𝑀⟩}‘1) = 0)
48	46, 47	mp1i 13	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → ({⟨1, 0⟩, ⟨2, 𝑀⟩}‘1) = 0)
49	45, 48	syl5eq 2868	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑋‘1) = 0)
50	42	fveq1i 6671	. . . . . . . . 9 ⊢ (𝑌‘1) = ({⟨1, 0⟩, ⟨2, 𝑁⟩}‘1)
51		fvpr1g 6954	. . . . . . . . . 10 ⊢ ((1 ∈ V ∧ 0 ∈ V ∧ 1 ≠ 2) → ({⟨1, 0⟩, ⟨2, 𝑁⟩}‘1) = 0)
52	46, 51	mp1i 13	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → ({⟨1, 0⟩, ⟨2, 𝑁⟩}‘1) = 0)
53	50, 52	syl5eq 2868	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑌‘1) = 0)
54	49, 53	eqtr4d 2859	. . . . . . 7 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑋‘1) = (𝑌‘1))
55		simp3 1134	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → 𝑀 ≠ 𝑁)
56	24	fveq1i 6671	. . . . . . . . 9 ⊢ (𝑋‘2) = ({⟨1, 0⟩, ⟨2, 𝑀⟩}‘2)
57		simp1 1132	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → 𝑀 ∈ ℝ)
58	8	a1i 11	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → 1 ≠ 2)
59		fvpr2g 6955	. . . . . . . . . 10 ⊢ ((2 ∈ V ∧ 𝑀 ∈ ℝ ∧ 1 ≠ 2) → ({⟨1, 0⟩, ⟨2, 𝑀⟩}‘2) = 𝑀)
60	4, 57, 58, 59	mp3an2i 1462	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → ({⟨1, 0⟩, ⟨2, 𝑀⟩}‘2) = 𝑀)
61	56, 60	syl5eq 2868	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑋‘2) = 𝑀)
62	42	fveq1i 6671	. . . . . . . . 9 ⊢ (𝑌‘2) = ({⟨1, 0⟩, ⟨2, 𝑁⟩}‘2)
63		simp2 1133	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → 𝑁 ∈ ℝ)
64		fvpr2g 6955	. . . . . . . . . 10 ⊢ ((2 ∈ V ∧ 𝑁 ∈ ℝ ∧ 1 ≠ 2) → ({⟨1, 0⟩, ⟨2, 𝑁⟩}‘2) = 𝑁)
65	4, 63, 58, 64	mp3an2i 1462	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → ({⟨1, 0⟩, ⟨2, 𝑁⟩}‘2) = 𝑁)
66	62, 65	syl5eq 2868	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑌‘2) = 𝑁)
67	55, 61, 66	3netr4d 3093	. . . . . . 7 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑋‘2) ≠ (𝑌‘2))
68	54, 67	jca 514	. . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) ≠ (𝑌‘2)))
69	27, 44, 68	3jca 1124	. . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) ≠ (𝑌‘2))))
70	69	adantl 484	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) → (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) ≠ (𝑌‘2))))
71		line2.e	. . . . 5 ⊢ 𝐸 = (ℝ^‘𝐼)
72		line2.l	. . . . 5 ⊢ 𝐿 = (Line_M‘𝐸)
73	16, 71, 25, 72	rrx2vlinest 44748	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) ≠ (𝑌‘2))) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ (𝑝‘1) = (𝑋‘1)})
74	70, 73	syl 17	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ (𝑝‘1) = (𝑋‘1)})
75	2, 74	eqeq12d 2837	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) → (𝐺 = (𝑋𝐿𝑌) ↔ {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶} = {𝑝 ∈ 𝑃 ∣ (𝑝‘1) = (𝑋‘1)}))
76	46, 47	ax-mp 5	. . . . . . 7 ⊢ ({⟨1, 0⟩, ⟨2, 𝑀⟩}‘1) = 0
77	45, 76	eqtri 2844	. . . . . 6 ⊢ (𝑋‘1) = 0
78	77	a1i 11	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) → (𝑋‘1) = 0)
79	78	eqeq2d 2832	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) → ((𝑝‘1) = (𝑋‘1) ↔ (𝑝‘1) = 0))
80	79	rabbidv 3480	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) → {𝑝 ∈ 𝑃 ∣ (𝑝‘1) = (𝑋‘1)} = {𝑝 ∈ 𝑃 ∣ (𝑝‘1) = 0})
81	80	eqeq2d 2832	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) → ({𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶} = {𝑝 ∈ 𝑃 ∣ (𝑝‘1) = (𝑋‘1)} ↔ {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶} = {𝑝 ∈ 𝑃 ∣ (𝑝‘1) = 0}))
82		rabbi 3383	. . 3 ⊢ (∀𝑝 ∈ 𝑃 (((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶 ↔ (𝑝‘1) = 0) ↔ {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶} = {𝑝 ∈ 𝑃 ∣ (𝑝‘1) = 0})
83	16, 25	line2ylem 44758	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (∀𝑝 ∈ 𝑃 (((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶 ↔ (𝑝‘1) = 0) → (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)))
84	83	adantr 483	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) → (∀𝑝 ∈ 𝑃 (((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶 ↔ (𝑝‘1) = 0) → (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)))
85		oveq1 7163	. . . . . . . . . . 11 ⊢ (𝐵 = 0 → (𝐵 · (𝑝‘2)) = (0 · (𝑝‘2)))
86	85	3ad2ant2 1130	. . . . . . . . . 10 ⊢ ((𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0) → (𝐵 · (𝑝‘2)) = (0 · (𝑝‘2)))
87	86	oveq2d 7172	. . . . . . . . 9 ⊢ ((𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0) → ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = ((𝐴 · (𝑝‘1)) + (0 · (𝑝‘2))))
88		simp3 1134	. . . . . . . . 9 ⊢ ((𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0) → 𝐶 = 0)
89	87, 88	eqeq12d 2837	. . . . . . . 8 ⊢ ((𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0) → (((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶 ↔ ((𝐴 · (𝑝‘1)) + (0 · (𝑝‘2))) = 0))
90	89	ad2antlr 725	. . . . . . 7 ⊢ (((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) ∧ (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)) ∧ 𝑝 ∈ 𝑃) → (((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶 ↔ ((𝐴 · (𝑝‘1)) + (0 · (𝑝‘2))) = 0))
91	16, 25	rrx2pyel 44719	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘2) ∈ ℝ)
92	91	recnd 10669	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘2) ∈ ℂ)
93	92	mul02d 10838	. . . . . . . . . . 11 ⊢ (𝑝 ∈ 𝑃 → (0 · (𝑝‘2)) = 0)
94	93	adantl 484	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) ∧ (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)) ∧ 𝑝 ∈ 𝑃) → (0 · (𝑝‘2)) = 0)
95	94	oveq2d 7172	. . . . . . . . 9 ⊢ (((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) ∧ (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)) ∧ 𝑝 ∈ 𝑃) → ((𝐴 · (𝑝‘1)) + (0 · (𝑝‘2))) = ((𝐴 · (𝑝‘1)) + 0))
96		simp1 1132	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℝ)
97	96	recnd 10669	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℂ)
98	97	ad3antrrr 728	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) ∧ (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)) ∧ 𝑝 ∈ 𝑃) → 𝐴 ∈ ℂ)
99	16, 25	rrx2pxel 44718	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘1) ∈ ℝ)
100	99	recnd 10669	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘1) ∈ ℂ)
101	100	adantl 484	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) ∧ (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)) ∧ 𝑝 ∈ 𝑃) → (𝑝‘1) ∈ ℂ)
102	98, 101	mulcld 10661	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) ∧ (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)) ∧ 𝑝 ∈ 𝑃) → (𝐴 · (𝑝‘1)) ∈ ℂ)
103	102	addid1d 10840	. . . . . . . . 9 ⊢ (((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) ∧ (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)) ∧ 𝑝 ∈ 𝑃) → ((𝐴 · (𝑝‘1)) + 0) = (𝐴 · (𝑝‘1)))
104	95, 103	eqtrd 2856	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) ∧ (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)) ∧ 𝑝 ∈ 𝑃) → ((𝐴 · (𝑝‘1)) + (0 · (𝑝‘2))) = (𝐴 · (𝑝‘1)))
105	104	eqeq1d 2823	. . . . . . 7 ⊢ (((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) ∧ (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)) ∧ 𝑝 ∈ 𝑃) → (((𝐴 · (𝑝‘1)) + (0 · (𝑝‘2))) = 0 ↔ (𝐴 · (𝑝‘1)) = 0))
106	98, 101	mul0ord 11290	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) ∧ (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)) ∧ 𝑝 ∈ 𝑃) → ((𝐴 · (𝑝‘1)) = 0 ↔ (𝐴 = 0 ∨ (𝑝‘1) = 0)))
107		eqneqall 3027	. . . . . . . . . . . . 13 ⊢ (𝐴 = 0 → (𝐴 ≠ 0 → (𝑝‘1) = 0))
108	107	com12 32	. . . . . . . . . . . 12 ⊢ (𝐴 ≠ 0 → (𝐴 = 0 → (𝑝‘1) = 0))
109	108	3ad2ant1 1129	. . . . . . . . . . 11 ⊢ ((𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0) → (𝐴 = 0 → (𝑝‘1) = 0))
110	109	ad2antlr 725	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) ∧ (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)) ∧ 𝑝 ∈ 𝑃) → (𝐴 = 0 → (𝑝‘1) = 0))
111		idd 24	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) ∧ (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)) ∧ 𝑝 ∈ 𝑃) → ((𝑝‘1) = 0 → (𝑝‘1) = 0))
112	110, 111	jaod 855	. . . . . . . . 9 ⊢ (((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) ∧ (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)) ∧ 𝑝 ∈ 𝑃) → ((𝐴 = 0 ∨ (𝑝‘1) = 0) → (𝑝‘1) = 0))
113		olc 864	. . . . . . . . 9 ⊢ ((𝑝‘1) = 0 → (𝐴 = 0 ∨ (𝑝‘1) = 0))
114	112, 113	impbid1 227	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) ∧ (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)) ∧ 𝑝 ∈ 𝑃) → ((𝐴 = 0 ∨ (𝑝‘1) = 0) ↔ (𝑝‘1) = 0))
115	106, 114	bitrd 281	. . . . . . 7 ⊢ (((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) ∧ (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)) ∧ 𝑝 ∈ 𝑃) → ((𝐴 · (𝑝‘1)) = 0 ↔ (𝑝‘1) = 0))
116	90, 105, 115	3bitrd 307	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) ∧ (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)) ∧ 𝑝 ∈ 𝑃) → (((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶 ↔ (𝑝‘1) = 0))
117	116	ralrimiva 3182	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) ∧ (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)) → ∀𝑝 ∈ 𝑃 (((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶 ↔ (𝑝‘1) = 0))
118	117	ex 415	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) → ((𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0) → ∀𝑝 ∈ 𝑃 (((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶 ↔ (𝑝‘1) = 0)))
119	84, 118	impbid 214	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) → (∀𝑝 ∈ 𝑃 (((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶 ↔ (𝑝‘1) = 0) ↔ (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)))
120	82, 119	syl5bbr 287	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) → ({𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶} = {𝑝 ∈ 𝑃 ∣ (𝑝‘1) = 0} ↔ (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)))
121	75, 81, 120	3bitrd 307	1 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) → (𝐺 = (𝑋𝐿𝑌) ↔ (𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 {crab 3142 Vcvv 3494 ⊆ wss 3936 {cpr 4569 ⟨cop 4573 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ↑_m cmap 8406 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 2c2 11693 ℝ^crrx 23986 Line_Mcline 44734

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-rp 12391 df-fz 12894 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-0g 16715 df-prds 16721 df-pws 16723 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-ghm 18356 df-cmn 18908 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-rnghom 19467 df-drng 19504 df-field 19505 df-subrg 19533 df-staf 19616 df-srng 19617 df-lmod 19636 df-lss 19704 df-sra 19944 df-rgmod 19945 df-cnfld 20546 df-refld 20749 df-dsmm 20876 df-frlm 20891 df-tng 23194 df-tcph 23773 df-rrx 23988 df-line 44736

This theorem is referenced by: (None)

W3C validator