line2y - Mathbox for Alexander van der Vekens

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Theorem line2y 49253

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 11138	. . . . . . . . . . . 12 ⊢ 1 ∈ V
4		2ex 12256	. . . . . . . . . . . 12 ⊢ 2 ∈ V
5	3, 4	pm3.2i 471	. . . . . . . . . . 11 ⊢ (1 ∈ V ∧ 2 ∈ V)
6		c0ex 11136	. . . . . . . . . . . 12 ⊢ 0 ∈ V
7	6	jctl 528	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℝ → (0 ∈ V ∧ 𝑀 ∈ ℝ))
8		1ne2 12382	. . . . . . . . . . . 12 ⊢ 1 ≠ 2
9	8	a1i 11	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℝ → 1 ≠ 2)
10		fprg 7105	. . . . . . . . . . . 12 ⊢ (((1 ∈ V ∧ 2 ∈ V) ∧ (0 ∈ V ∧ 𝑀 ∈ ℝ) ∧ 1 ≠ 2) → {⟨1, 0⟩, ⟨2, 𝑀⟩}:{1, 2}⟶{0, 𝑀})
11		0red 11145	. . . . . . . . . . . . 13 ⊢ (((1 ∈ V ∧ 2 ∈ V) ∧ (0 ∈ V ∧ 𝑀 ∈ ℝ) ∧ 1 ≠ 2) → 0 ∈ ℝ)
12		simp2r 1207	. . . . . . . . . . . . 13 ⊢ (((1 ∈ V ∧ 2 ∈ V) ∧ (0 ∈ V ∧ 𝑀 ∈ ℝ) ∧ 1 ≠ 2) → 𝑀 ∈ ℝ)
13	11, 12	prssd 4760	. . . . . . . . . . . 12 ⊢ (((1 ∈ V ∧ 2 ∈ V) ∧ (0 ∈ V ∧ 𝑀 ∈ ℝ) ∧ 1 ≠ 2) → {0, 𝑀} ⊆ ℝ)
14	10, 13	fssd 6679	. . . . . . . . . . 11 ⊢ (((1 ∈ V ∧ 2 ∈ V) ∧ (0 ∈ V ∧ 𝑀 ∈ ℝ) ∧ 1 ≠ 2) → {⟨1, 0⟩, ⟨2, 𝑀⟩}:{1, 2}⟶ℝ)
15	5, 7, 9, 14	mp3an2i 1474	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℝ → {⟨1, 0⟩, ⟨2, 𝑀⟩}:{1, 2}⟶ℝ)
16		line2.i	. . . . . . . . . . 11 ⊢ 𝐼 = {1, 2}
17	16	feq2i 6654	. . . . . . . . . 10 ⊢ ({⟨1, 0⟩, ⟨2, 𝑀⟩}:𝐼⟶ℝ ↔ {⟨1, 0⟩, ⟨2, 𝑀⟩}:{1, 2}⟶ℝ)
18	15, 17	sylibr 235	. . . . . . . . 9 ⊢ (𝑀 ∈ ℝ → {⟨1, 0⟩, ⟨2, 𝑀⟩}:𝐼⟶ℝ)
19		reex 11127	. . . . . . . . . 10 ⊢ ℝ ∈ V
20		prex 5374	. . . . . . . . . . 11 ⊢ {1, 2} ∈ V
21	16, 20	eqeltri 2836	. . . . . . . . . 10 ⊢ 𝐼 ∈ V
22	19, 21	elmap 8816	. . . . . . . . 9 ⊢ ({⟨1, 0⟩, ⟨2, 𝑀⟩} ∈ (ℝ ↑_m 𝐼) ↔ {⟨1, 0⟩, ⟨2, 𝑀⟩}:𝐼⟶ℝ)
23	18, 22	sylibr 235	. . . . . . . 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 2857	. . . . . . 7 ⊢ (𝑀 ∈ ℝ → 𝑋 ∈ 𝑃)
27	26	3ad2ant1 1139	. . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → 𝑋 ∈ 𝑃)
28	6	jctl 528	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → (0 ∈ V ∧ 𝑁 ∈ ℝ))
29	8	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → 1 ≠ 2)
30		fprg 7105	. . . . . . . . . . . 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 1474	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℝ → {⟨1, 0⟩, ⟨2, 𝑁⟩}:{1, 2}⟶{0, 𝑁})
32		0re 11144	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
33		prssi 4759	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ) → {0, 𝑁} ⊆ ℝ)
34	32, 33	mpan 696	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℝ → {0, 𝑁} ⊆ ℝ)
35	31, 34	fssd 6679	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ → {⟨1, 0⟩, ⟨2, 𝑁⟩}:{1, 2}⟶ℝ)
36	16	feq2i 6654	. . . . . . . . . 10 ⊢ ({⟨1, 0⟩, ⟨2, 𝑁⟩}:𝐼⟶ℝ ↔ {⟨1, 0⟩, ⟨2, 𝑁⟩}:{1, 2}⟶ℝ)
37	35, 36	sylibr 235	. . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → {⟨1, 0⟩, ⟨2, 𝑁⟩}:𝐼⟶ℝ)
38	19, 21	pm3.2i 471	. . . . . . . . . 10 ⊢ (ℝ ∈ V ∧ 𝐼 ∈ V)
39		elmapg 8783	. . . . . . . . . 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 258	. . . . . . . 8 ⊢ (𝑁 ∈ ℝ → {⟨1, 0⟩, ⟨2, 𝑁⟩} ∈ (ℝ ↑_m 𝐼))
42		line2y.y	. . . . . . . 8 ⊢ 𝑌 = {⟨1, 0⟩, ⟨2, 𝑁⟩}
43	41, 42, 25	3eltr4g 2857	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → 𝑌 ∈ 𝑃)
44	43	3ad2ant2 1140	. . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → 𝑌 ∈ 𝑃)
45	24	fveq1i 6835	. . . . . . . . 9 ⊢ (𝑋‘1) = ({⟨1, 0⟩, ⟨2, 𝑀⟩}‘1)
46	3, 6, 8	3pm3.2i 1346	. . . . . . . . . 10 ⊢ (1 ∈ V ∧ 0 ∈ V ∧ 1 ≠ 2)
47		fvpr1g 7141	. . . . . . . . . 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 2787	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑋‘1) = 0)
50	42	fveq1i 6835	. . . . . . . . 9 ⊢ (𝑌‘1) = ({⟨1, 0⟩, ⟨2, 𝑁⟩}‘1)
51		fvpr1g 7141	. . . . . . . . . 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 2787	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑌‘1) = 0)
54	49, 53	eqtr4d 2778	. . . . . . 7 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑋‘1) = (𝑌‘1))
55		simp3 1144	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → 𝑀 ≠ 𝑁)
56	24	fveq1i 6835	. . . . . . . . 9 ⊢ (𝑋‘2) = ({⟨1, 0⟩, ⟨2, 𝑀⟩}‘2)
57		simp1 1142	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → 𝑀 ∈ ℝ)
58	8	a1i 11	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → 1 ≠ 2)
59		fvpr2g 7142	. . . . . . . . . 10 ⊢ ((2 ∈ V ∧ 𝑀 ∈ ℝ ∧ 1 ≠ 2) → ({⟨1, 0⟩, ⟨2, 𝑀⟩}‘2) = 𝑀)
60	4, 57, 58, 59	mp3an2i 1474	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → ({⟨1, 0⟩, ⟨2, 𝑀⟩}‘2) = 𝑀)
61	56, 60	eqtrid 2787	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑋‘2) = 𝑀)
62	42	fveq1i 6835	. . . . . . . . 9 ⊢ (𝑌‘2) = ({⟨1, 0⟩, ⟨2, 𝑁⟩}‘2)
63		simp2 1143	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → 𝑁 ∈ ℝ)
64		fvpr2g 7142	. . . . . . . . . 10 ⊢ ((2 ∈ V ∧ 𝑁 ∈ ℝ ∧ 1 ≠ 2) → ({⟨1, 0⟩, ⟨2, 𝑁⟩}‘2) = 𝑁)
65	4, 63, 58, 64	mp3an2i 1474	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → ({⟨1, 0⟩, ⟨2, 𝑁⟩}‘2) = 𝑁)
66	62, 65	eqtrid 2787	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑌‘2) = 𝑁)
67	55, 61, 66	3netr4d 3012	. . . . . . 7 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑋‘2) ≠ (𝑌‘2))
68	54, 67	jca 516	. . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) ≠ (𝑌‘2)))
69	27, 44, 68	3jca 1134	. . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ ((𝑋‘1) = (𝑌‘1) ∧ (𝑋‘2) ≠ (𝑌‘2))))
70	69	adantl 482	. . . 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 49239	. . . 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 2756	. 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 2763	. . . . . 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 2751	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) → ((𝑝‘1) = (𝑋‘1) ↔ (𝑝‘1) = 0))
80	79	rabbidv 3399	. . 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 2751	. 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 3422	. . 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 49249	. . . . 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 481	. . . 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 7370	. . . . . . . . . . 11 ⊢ (𝐵 = 0 → (𝐵 · (𝑝‘2)) = (0 · (𝑝‘2)))
86	85	3ad2ant2 1140	. . . . . . . . . 10 ⊢ ((𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0) → (𝐵 · (𝑝‘2)) = (0 · (𝑝‘2)))
87	86	oveq2d 7379	. . . . . . . . 9 ⊢ ((𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0) → ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = ((𝐴 · (𝑝‘1)) + (0 · (𝑝‘2))))
88		simp3 1144	. . . . . . . . 9 ⊢ ((𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0) → 𝐶 = 0)
89	87, 88	eqeq12d 2756	. . . . . . . 8 ⊢ ((𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0) → (((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶 ↔ ((𝐴 · (𝑝‘1)) + (0 · (𝑝‘2))) = 0))
90	89	ad2antlr 733	. . . . . . 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 49210	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘2) ∈ ℝ)
92	91	recnd 11171	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘2) ∈ ℂ)
93	92	mul02d 11342	. . . . . . . . . . 11 ⊢ (𝑝 ∈ 𝑃 → (0 · (𝑝‘2)) = 0)
94	93	adantl 482	. . . . . . . . . 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 7379	. . . . . . . . 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 1142	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℝ)
97	96	recnd 11171	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℂ)
98	97	ad3antrrr 736	. . . . . . . . . . 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 49209	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘1) ∈ ℝ)
100	99	recnd 11171	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘1) ∈ ℂ)
101	100	adantl 482	. . . . . . . . . . 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 11163	. . . . . . . . . 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 11344	. . . . . . . . 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 2775	. . . . . . . 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 2742	. . . . . . 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 11796	. . . . . . . 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 2946	. . . . . . . . . . . . 13 ⊢ (𝐴 = 0 → (𝐴 ≠ 0 → (𝑝‘1) = 0))
108	107	com12 32	. . . . . . . . . . . 12 ⊢ (𝐴 ≠ 0 → (𝐴 = 0 → (𝑝‘1) = 0))
109	108	3ad2ant1 1139	. . . . . . . . . . 11 ⊢ ((𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0) → (𝐴 = 0 → (𝑝‘1) = 0))
110	109	ad2antlr 733	. . . . . . . . . 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 865	. . . . . . . . 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 874	. . . . . . . . 9 ⊢ ((𝑝‘1) = 0 → (𝐴 = 0 ∨ (𝑝‘1) = 0))
114	112, 113	impbid1 226	. . . . . . . 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 280	. . . . . . 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 306	. . . . . 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 3132	. . . . 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 413	. . . 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 213	. . 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 286	. 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 306	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 207 ∧ wa 396 ∨ wo 853 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ≠ wne 2935 ∀wral 3054 {crab 3392 Vcvv 3432 ⊆ wss 3890 {cpr 4564 ⟨cop 4568 ⟶wf 6488 ‘cfv 6492 (class class class)co 7363 ↑_m cmap 8770 ℂcc 11034 ℝcr 11035 0cc0 11036 1c1 11037 + caddc 11039 · cmul 11041 2c2 12234 ℝ^crrx 25375 Line_Mcline 49225

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114 ax-addf 11115 ax-mulf 11116

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-tpos 8173 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-sup 9352 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-rp 12941 df-fz 13460 df-seq 13962 df-exp 14022 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-0g 17402 df-prds 17408 df-pws 17410 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-mhm 18749 df-grp 18910 df-minusg 18911 df-sbg 18912 df-subg 19097 df-ghm 19186 df-cmn 19755 df-abl 19756 df-mgp 20120 df-rng 20132 df-ur 20161 df-ring 20214 df-cring 20215 df-oppr 20315 df-dvdsr 20335 df-unit 20336 df-invr 20366 df-dvr 20379 df-rhm 20450 df-subrng 20525 df-subrg 20549 df-drng 20710 df-field 20711 df-staf 20818 df-srng 20819 df-lmod 20859 df-lss 20929 df-sra 21170 df-rgmod 21171 df-cnfld 21355 df-refld 21587 df-dsmm 21714 df-frlm 21729 df-tng 24574 df-tcph 25161 df-rrx 25377 df-line 49227

This theorem is referenced by: (None)

W3C validator