line2y - Mathbox for Alexander van der Vekens

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Theorem line2y 48881

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 11115	. . . . . . . . . . . 12 ⊢ 1 ∈ V
4		2ex 12209	. . . . . . . . . . . 12 ⊢ 2 ∈ V
5	3, 4	pm3.2i 470	. . . . . . . . . . 11 ⊢ (1 ∈ V ∧ 2 ∈ V)
6		c0ex 11113	. . . . . . . . . . . 12 ⊢ 0 ∈ V
7	6	jctl 523	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℝ → (0 ∈ V ∧ 𝑀 ∈ ℝ))
8		1ne2 12335	. . . . . . . . . . . 12 ⊢ 1 ≠ 2
9	8	a1i 11	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℝ → 1 ≠ 2)
10		fprg 7094	. . . . . . . . . . . 12 ⊢ (((1 ∈ V ∧ 2 ∈ V) ∧ (0 ∈ V ∧ 𝑀 ∈ ℝ) ∧ 1 ≠ 2) → {⟨1, 0⟩, ⟨2, 𝑀⟩}:{1, 2}⟶{0, 𝑀})
11		0red 11122	. . . . . . . . . . . . 13 ⊢ (((1 ∈ V ∧ 2 ∈ V) ∧ (0 ∈ V ∧ 𝑀 ∈ ℝ) ∧ 1 ≠ 2) → 0 ∈ ℝ)
12		simp2r 1201	. . . . . . . . . . . . 13 ⊢ (((1 ∈ V ∧ 2 ∈ V) ∧ (0 ∈ V ∧ 𝑀 ∈ ℝ) ∧ 1 ≠ 2) → 𝑀 ∈ ℝ)
13	11, 12	prssd 4773	. . . . . . . . . . . 12 ⊢ (((1 ∈ V ∧ 2 ∈ V) ∧ (0 ∈ V ∧ 𝑀 ∈ ℝ) ∧ 1 ≠ 2) → {0, 𝑀} ⊆ ℝ)
14	10, 13	fssd 6673	. . . . . . . . . . 11 ⊢ (((1 ∈ V ∧ 2 ∈ V) ∧ (0 ∈ V ∧ 𝑀 ∈ ℝ) ∧ 1 ≠ 2) → {⟨1, 0⟩, ⟨2, 𝑀⟩}:{1, 2}⟶ℝ)
15	5, 7, 9, 14	mp3an2i 1468	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℝ → {⟨1, 0⟩, ⟨2, 𝑀⟩}:{1, 2}⟶ℝ)
16		line2.i	. . . . . . . . . . 11 ⊢ 𝐼 = {1, 2}
17	16	feq2i 6648	. . . . . . . . . 10 ⊢ ({⟨1, 0⟩, ⟨2, 𝑀⟩}:𝐼⟶ℝ ↔ {⟨1, 0⟩, ⟨2, 𝑀⟩}:{1, 2}⟶ℝ)
18	15, 17	sylibr 234	. . . . . . . . 9 ⊢ (𝑀 ∈ ℝ → {⟨1, 0⟩, ⟨2, 𝑀⟩}:𝐼⟶ℝ)
19		reex 11104	. . . . . . . . . 10 ⊢ ℝ ∈ V
20		prex 5377	. . . . . . . . . . 11 ⊢ {1, 2} ∈ V
21	16, 20	eqeltri 2829	. . . . . . . . . 10 ⊢ 𝐼 ∈ V
22	19, 21	elmap 8801	. . . . . . . . 9 ⊢ ({⟨1, 0⟩, ⟨2, 𝑀⟩} ∈ (ℝ ↑_m 𝐼) ↔ {⟨1, 0⟩, ⟨2, 𝑀⟩}:𝐼⟶ℝ)
23	18, 22	sylibr 234	. . . . . . . 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 2850	. . . . . . 7 ⊢ (𝑀 ∈ ℝ → 𝑋 ∈ 𝑃)
27	26	3ad2ant1 1133	. . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → 𝑋 ∈ 𝑃)
28	6	jctl 523	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → (0 ∈ V ∧ 𝑁 ∈ ℝ))
29	8	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → 1 ≠ 2)
30		fprg 7094	. . . . . . . . . . . 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 1468	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℝ → {⟨1, 0⟩, ⟨2, 𝑁⟩}:{1, 2}⟶{0, 𝑁})
32		0re 11121	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
33		prssi 4772	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ) → {0, 𝑁} ⊆ ℝ)
34	32, 33	mpan 690	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℝ → {0, 𝑁} ⊆ ℝ)
35	31, 34	fssd 6673	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ → {⟨1, 0⟩, ⟨2, 𝑁⟩}:{1, 2}⟶ℝ)
36	16	feq2i 6648	. . . . . . . . . 10 ⊢ ({⟨1, 0⟩, ⟨2, 𝑁⟩}:𝐼⟶ℝ ↔ {⟨1, 0⟩, ⟨2, 𝑁⟩}:{1, 2}⟶ℝ)
37	35, 36	sylibr 234	. . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → {⟨1, 0⟩, ⟨2, 𝑁⟩}:𝐼⟶ℝ)
38	19, 21	pm3.2i 470	. . . . . . . . . 10 ⊢ (ℝ ∈ V ∧ 𝐼 ∈ V)
39		elmapg 8769	. . . . . . . . . 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 257	. . . . . . . 8 ⊢ (𝑁 ∈ ℝ → {⟨1, 0⟩, ⟨2, 𝑁⟩} ∈ (ℝ ↑_m 𝐼))
42		line2y.y	. . . . . . . 8 ⊢ 𝑌 = {⟨1, 0⟩, ⟨2, 𝑁⟩}
43	41, 42, 25	3eltr4g 2850	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → 𝑌 ∈ 𝑃)
44	43	3ad2ant2 1134	. . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → 𝑌 ∈ 𝑃)
45	24	fveq1i 6829	. . . . . . . . 9 ⊢ (𝑋‘1) = ({⟨1, 0⟩, ⟨2, 𝑀⟩}‘1)
46	3, 6, 8	3pm3.2i 1340	. . . . . . . . . 10 ⊢ (1 ∈ V ∧ 0 ∈ V ∧ 1 ≠ 2)
47		fvpr1g 7130	. . . . . . . . . 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	eqtrid 2780	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑋‘1) = 0)
50	42	fveq1i 6829	. . . . . . . . 9 ⊢ (𝑌‘1) = ({⟨1, 0⟩, ⟨2, 𝑁⟩}‘1)
51		fvpr1g 7130	. . . . . . . . . 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	eqtrid 2780	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑌‘1) = 0)
54	49, 53	eqtr4d 2771	. . . . . . 7 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑋‘1) = (𝑌‘1))
55		simp3 1138	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → 𝑀 ≠ 𝑁)
56	24	fveq1i 6829	. . . . . . . . 9 ⊢ (𝑋‘2) = ({⟨1, 0⟩, ⟨2, 𝑀⟩}‘2)
57		simp1 1136	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → 𝑀 ∈ ℝ)
58	8	a1i 11	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → 1 ≠ 2)
59		fvpr2g 7131	. . . . . . . . . 10 ⊢ ((2 ∈ V ∧ 𝑀 ∈ ℝ ∧ 1 ≠ 2) → ({⟨1, 0⟩, ⟨2, 𝑀⟩}‘2) = 𝑀)
60	4, 57, 58, 59	mp3an2i 1468	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → ({⟨1, 0⟩, ⟨2, 𝑀⟩}‘2) = 𝑀)
61	56, 60	eqtrid 2780	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑋‘2) = 𝑀)
62	42	fveq1i 6829	. . . . . . . . 9 ⊢ (𝑌‘2) = ({⟨1, 0⟩, ⟨2, 𝑁⟩}‘2)
63		simp2 1137	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → 𝑁 ∈ ℝ)
64		fvpr2g 7131	. . . . . . . . . 10 ⊢ ((2 ∈ V ∧ 𝑁 ∈ ℝ ∧ 1 ≠ 2) → ({⟨1, 0⟩, ⟨2, 𝑁⟩}‘2) = 𝑁)
65	4, 63, 58, 64	mp3an2i 1468	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → ({⟨1, 0⟩, ⟨2, 𝑁⟩}‘2) = 𝑁)
66	62, 65	eqtrid 2780	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑌‘2) = 𝑁)
67	55, 61, 66	3netr4d 3006	. . . . . . 7 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑋‘2) ≠ (𝑌‘2))
68	54, 67	jca 511	. . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) ≠ (𝑌‘2)))
69	27, 44, 68	3jca 1128	. . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) ≠ (𝑌‘2))))
70	69	adantl 481	. . . 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 48867	. . . 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 2749	. 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 2756	. . . . . 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 2744	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) → ((𝑝‘1) = (𝑋‘1) ↔ (𝑝‘1) = 0))
80	79	rabbidv 3403	. . 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 2744	. 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 3426	. . 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 48877	. . . . 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 480	. . . 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 7359	. . . . . . . . . . 11 ⊢ (𝐵 = 0 → (𝐵 · (𝑝‘2)) = (0 · (𝑝‘2)))
86	85	3ad2ant2 1134	. . . . . . . . . 10 ⊢ ((𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0) → (𝐵 · (𝑝‘2)) = (0 · (𝑝‘2)))
87	86	oveq2d 7368	. . . . . . . . 9 ⊢ ((𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0) → ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = ((𝐴 · (𝑝‘1)) + (0 · (𝑝‘2))))
88		simp3 1138	. . . . . . . . 9 ⊢ ((𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0) → 𝐶 = 0)
89	87, 88	eqeq12d 2749	. . . . . . . 8 ⊢ ((𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0) → (((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶 ↔ ((𝐴 · (𝑝‘1)) + (0 · (𝑝‘2))) = 0))
90	89	ad2antlr 727	. . . . . . 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 48838	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘2) ∈ ℝ)
92	91	recnd 11147	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘2) ∈ ℂ)
93	92	mul02d 11318	. . . . . . . . . . 11 ⊢ (𝑝 ∈ 𝑃 → (0 · (𝑝‘2)) = 0)
94	93	adantl 481	. . . . . . . . . 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 7368	. . . . . . . . 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 1136	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℝ)
97	96	recnd 11147	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℂ)
98	97	ad3antrrr 730	. . . . . . . . . . 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 48837	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘1) ∈ ℝ)
100	99	recnd 11147	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘1) ∈ ℂ)
101	100	adantl 481	. . . . . . . . . . 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 11139	. . . . . . . . . 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	addridd 11320	. . . . . . . . 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 2768	. . . . . . . 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 2735	. . . . . . 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 11772	. . . . . . . 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 2940	. . . . . . . . . . . . 13 ⊢ (𝐴 = 0 → (𝐴 ≠ 0 → (𝑝‘1) = 0))
108	107	com12 32	. . . . . . . . . . . 12 ⊢ (𝐴 ≠ 0 → (𝐴 = 0 → (𝑝‘1) = 0))
109	108	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0) → (𝐴 = 0 → (𝑝‘1) = 0))
110	109	ad2antlr 727	. . . . . . . . . 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 859	. . . . . . . . 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 868	. . . . . . . . 9 ⊢ ((𝑝‘1) = 0 → (𝐴 = 0 ∨ (𝑝‘1) = 0))
114	112, 113	impbid1 225	. . . . . . . 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 279	. . . . . . 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 305	. . . . . 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 3125	. . . . 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 412	. . . 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 212	. . 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	bitr3id 285	. 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 305	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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2929 ∀wral 3048 {crab 3396 Vcvv 3437 ⊆ wss 3898 {cpr 4577 ⟨cop 4581 ⟶wf 6482 ‘cfv 6486 (class class class)co 7352 ↑_m cmap 8756 ℂcc 11011 ℝcr 11012 0cc0 11013 1c1 11014 + caddc 11016 · cmul 11018 2c2 12187 ℝ^crrx 25311 Line_Mcline 48853

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-pre-sup 11091 ax-addf 11092 ax-mulf 11093

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-tpos 8162 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-er 8628 df-map 8758 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9253 df-sup 9333 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-5 12198 df-6 12199 df-7 12200 df-8 12201 df-9 12202 df-n0 12389 df-z 12476 df-dec 12595 df-uz 12739 df-rp 12893 df-fz 13410 df-seq 13911 df-exp 13971 df-cj 15008 df-re 15009 df-im 15010 df-sqrt 15144 df-abs 15145 df-struct 17060 df-sets 17077 df-slot 17095 df-ndx 17107 df-base 17123 df-ress 17144 df-plusg 17176 df-mulr 17177 df-starv 17178 df-sca 17179 df-vsca 17180 df-ip 17181 df-tset 17182 df-ple 17183 df-ds 17185 df-unif 17186 df-hom 17187 df-cco 17188 df-0g 17347 df-prds 17353 df-pws 17355 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-mhm 18693 df-grp 18851 df-minusg 18852 df-sbg 18853 df-subg 19038 df-ghm 19127 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-ring 20155 df-cring 20156 df-oppr 20257 df-dvdsr 20277 df-unit 20278 df-invr 20308 df-dvr 20321 df-rhm 20392 df-subrng 20463 df-subrg 20487 df-drng 20648 df-field 20649 df-staf 20756 df-srng 20757 df-lmod 20797 df-lss 20867 df-sra 21109 df-rgmod 21110 df-cnfld 21294 df-refld 21544 df-dsmm 21671 df-frlm 21686 df-tng 24500 df-tcph 25097 df-rrx 25313 df-line 48855

This theorem is referenced by: (None)

W3C validator