line2y - Mathbox for Alexander van der Vekens

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Theorem line2y 47603

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 11217	. . . . . . . . . . . 12 ⊢ 1 ∈ V
4		2ex 12296	. . . . . . . . . . . 12 ⊢ 2 ∈ V
5	3, 4	pm3.2i 470	. . . . . . . . . . 11 ⊢ (1 ∈ V ∧ 2 ∈ V)
6		c0ex 11215	. . . . . . . . . . . 12 ⊢ 0 ∈ V
7	6	jctl 523	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℝ → (0 ∈ V ∧ 𝑀 ∈ ℝ))
8		1ne2 12427	. . . . . . . . . . . 12 ⊢ 1 ≠ 2
9	8	a1i 11	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℝ → 1 ≠ 2)
10		fprg 7155	. . . . . . . . . . . 12 ⊢ (((1 ∈ V ∧ 2 ∈ V) ∧ (0 ∈ V ∧ 𝑀 ∈ ℝ) ∧ 1 ≠ 2) → {⟨1, 0⟩, ⟨2, 𝑀⟩}:{1, 2}⟶{0, 𝑀})
11		0red 11224	. . . . . . . . . . . . 13 ⊢ (((1 ∈ V ∧ 2 ∈ V) ∧ (0 ∈ V ∧ 𝑀 ∈ ℝ) ∧ 1 ≠ 2) → 0 ∈ ℝ)
12		simp2r 1199	. . . . . . . . . . . . 13 ⊢ (((1 ∈ V ∧ 2 ∈ V) ∧ (0 ∈ V ∧ 𝑀 ∈ ℝ) ∧ 1 ≠ 2) → 𝑀 ∈ ℝ)
13	11, 12	prssd 4825	. . . . . . . . . . . 12 ⊢ (((1 ∈ V ∧ 2 ∈ V) ∧ (0 ∈ V ∧ 𝑀 ∈ ℝ) ∧ 1 ≠ 2) → {0, 𝑀} ⊆ ℝ)
14	10, 13	fssd 6735	. . . . . . . . . . 11 ⊢ (((1 ∈ V ∧ 2 ∈ V) ∧ (0 ∈ V ∧ 𝑀 ∈ ℝ) ∧ 1 ≠ 2) → {⟨1, 0⟩, ⟨2, 𝑀⟩}:{1, 2}⟶ℝ)
15	5, 7, 9, 14	mp3an2i 1465	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℝ → {⟨1, 0⟩, ⟨2, 𝑀⟩}:{1, 2}⟶ℝ)
16		line2.i	. . . . . . . . . . 11 ⊢ 𝐼 = {1, 2}
17	16	feq2i 6709	. . . . . . . . . 10 ⊢ ({⟨1, 0⟩, ⟨2, 𝑀⟩}:𝐼⟶ℝ ↔ {⟨1, 0⟩, ⟨2, 𝑀⟩}:{1, 2}⟶ℝ)
18	15, 17	sylibr 233	. . . . . . . . 9 ⊢ (𝑀 ∈ ℝ → {⟨1, 0⟩, ⟨2, 𝑀⟩}:𝐼⟶ℝ)
19		reex 11207	. . . . . . . . . 10 ⊢ ℝ ∈ V
20		prex 5432	. . . . . . . . . . 11 ⊢ {1, 2} ∈ V
21	16, 20	eqeltri 2828	. . . . . . . . . 10 ⊢ 𝐼 ∈ V
22	19, 21	elmap 8871	. . . . . . . . 9 ⊢ ({⟨1, 0⟩, ⟨2, 𝑀⟩} ∈ (ℝ ↑_m 𝐼) ↔ {⟨1, 0⟩, ⟨2, 𝑀⟩}:𝐼⟶ℝ)
23	18, 22	sylibr 233	. . . . . . . 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 2849	. . . . . . 7 ⊢ (𝑀 ∈ ℝ → 𝑋 ∈ 𝑃)
27	26	3ad2ant1 1132	. . . . . 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 7155	. . . . . . . . . . . 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 1465	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℝ → {⟨1, 0⟩, ⟨2, 𝑁⟩}:{1, 2}⟶{0, 𝑁})
32		0re 11223	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
33		prssi 4824	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ) → {0, 𝑁} ⊆ ℝ)
34	32, 33	mpan 687	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℝ → {0, 𝑁} ⊆ ℝ)
35	31, 34	fssd 6735	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ → {⟨1, 0⟩, ⟨2, 𝑁⟩}:{1, 2}⟶ℝ)
36	16	feq2i 6709	. . . . . . . . . 10 ⊢ ({⟨1, 0⟩, ⟨2, 𝑁⟩}:𝐼⟶ℝ ↔ {⟨1, 0⟩, ⟨2, 𝑁⟩}:{1, 2}⟶ℝ)
37	35, 36	sylibr 233	. . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → {⟨1, 0⟩, ⟨2, 𝑁⟩}:𝐼⟶ℝ)
38	19, 21	pm3.2i 470	. . . . . . . . . 10 ⊢ (ℝ ∈ V ∧ 𝐼 ∈ V)
39		elmapg 8839	. . . . . . . . . 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 2849	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → 𝑌 ∈ 𝑃)
44	43	3ad2ant2 1133	. . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → 𝑌 ∈ 𝑃)
45	24	fveq1i 6892	. . . . . . . . 9 ⊢ (𝑋‘1) = ({⟨1, 0⟩, ⟨2, 𝑀⟩}‘1)
46	3, 6, 8	3pm3.2i 1338	. . . . . . . . . 10 ⊢ (1 ∈ V ∧ 0 ∈ V ∧ 1 ≠ 2)
47		fvpr1g 7190	. . . . . . . . . 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 2783	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑋‘1) = 0)
50	42	fveq1i 6892	. . . . . . . . 9 ⊢ (𝑌‘1) = ({⟨1, 0⟩, ⟨2, 𝑁⟩}‘1)
51		fvpr1g 7190	. . . . . . . . . 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 2783	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑌‘1) = 0)
54	49, 53	eqtr4d 2774	. . . . . . 7 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑋‘1) = (𝑌‘1))
55		simp3 1137	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → 𝑀 ≠ 𝑁)
56	24	fveq1i 6892	. . . . . . . . 9 ⊢ (𝑋‘2) = ({⟨1, 0⟩, ⟨2, 𝑀⟩}‘2)
57		simp1 1135	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → 𝑀 ∈ ℝ)
58	8	a1i 11	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → 1 ≠ 2)
59		fvpr2g 7191	. . . . . . . . . 10 ⊢ ((2 ∈ V ∧ 𝑀 ∈ ℝ ∧ 1 ≠ 2) → ({⟨1, 0⟩, ⟨2, 𝑀⟩}‘2) = 𝑀)
60	4, 57, 58, 59	mp3an2i 1465	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → ({⟨1, 0⟩, ⟨2, 𝑀⟩}‘2) = 𝑀)
61	56, 60	eqtrid 2783	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑋‘2) = 𝑀)
62	42	fveq1i 6892	. . . . . . . . 9 ⊢ (𝑌‘2) = ({⟨1, 0⟩, ⟨2, 𝑁⟩}‘2)
63		simp2 1136	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → 𝑁 ∈ ℝ)
64		fvpr2g 7191	. . . . . . . . . 10 ⊢ ((2 ∈ V ∧ 𝑁 ∈ ℝ ∧ 1 ≠ 2) → ({⟨1, 0⟩, ⟨2, 𝑁⟩}‘2) = 𝑁)
65	4, 63, 58, 64	mp3an2i 1465	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → ({⟨1, 0⟩, ⟨2, 𝑁⟩}‘2) = 𝑁)
66	62, 65	eqtrid 2783	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁) → (𝑌‘2) = 𝑁)
67	55, 61, 66	3netr4d 3017	. . . . . . 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 1127	. . . . 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 47589	. . . 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 2747	. 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 2759	. . . . . 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 2742	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ≠ 𝑁)) → ((𝑝‘1) = (𝑋‘1) ↔ (𝑝‘1) = 0))
80	79	rabbidv 3439	. . 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 2742	. 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 3461	. . 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 47599	. . . . 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 7419	. . . . . . . . . . 11 ⊢ (𝐵 = 0 → (𝐵 · (𝑝‘2)) = (0 · (𝑝‘2)))
86	85	3ad2ant2 1133	. . . . . . . . . 10 ⊢ ((𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0) → (𝐵 · (𝑝‘2)) = (0 · (𝑝‘2)))
87	86	oveq2d 7428	. . . . . . . . 9 ⊢ ((𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0) → ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = ((𝐴 · (𝑝‘1)) + (0 · (𝑝‘2))))
88		simp3 1137	. . . . . . . . 9 ⊢ ((𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0) → 𝐶 = 0)
89	87, 88	eqeq12d 2747	. . . . . . . 8 ⊢ ((𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0) → (((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶 ↔ ((𝐴 · (𝑝‘1)) + (0 · (𝑝‘2))) = 0))
90	89	ad2antlr 724	. . . . . . 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 47560	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘2) ∈ ℝ)
92	91	recnd 11249	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘2) ∈ ℂ)
93	92	mul02d 11419	. . . . . . . . . . 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 7428	. . . . . . . . 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 1135	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℝ)
97	96	recnd 11249	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℂ)
98	97	ad3antrrr 727	. . . . . . . . . . 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 47559	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘1) ∈ ℝ)
100	99	recnd 11249	. . . . . . . . . . . 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 11241	. . . . . . . . . 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 11421	. . . . . . . . 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 2771	. . . . . . . 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 2733	. . . . . . 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 11871	. . . . . . . 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 2950	. . . . . . . . . . . . 13 ⊢ (𝐴 = 0 → (𝐴 ≠ 0 → (𝑝‘1) = 0))
108	107	com12 32	. . . . . . . . . . . 12 ⊢ (𝐴 ≠ 0 → (𝐴 = 0 → (𝑝‘1) = 0))
109	108	3ad2ant1 1132	. . . . . . . . . . 11 ⊢ ((𝐴 ≠ 0 ∧ 𝐵 = 0 ∧ 𝐶 = 0) → (𝐴 = 0 → (𝑝‘1) = 0))
110	109	ad2antlr 724	. . . . . . . . . 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 856	. . . . . . . . 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 865	. . . . . . . . 9 ⊢ ((𝑝‘1) = 0 → (𝐴 = 0 ∨ (𝑝‘1) = 0))
114	112, 113	impbid1 224	. . . . . . . 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 3145	. . . . 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 211	. . 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 205 ∧ wa 395 ∨ wo 844 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 {crab 3431 Vcvv 3473 ⊆ wss 3948 {cpr 4630 ⟨cop 4634 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ↑_m cmap 8826 ℂcc 11114 ℝcr 11115 0cc0 11116 1c1 11117 + caddc 11119 · cmul 11121 2c2 12274 ℝ^crrx 25231 Line_Mcline 47575

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 ax-mulf 11196

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-sup 9443 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-rp 12982 df-fz 13492 df-seq 13974 df-exp 14035 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-0g 17394 df-prds 17400 df-pws 17402 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19046 df-ghm 19135 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-rhm 20370 df-subrng 20442 df-subrg 20467 df-drng 20585 df-field 20586 df-staf 20684 df-srng 20685 df-lmod 20704 df-lss 20775 df-sra 21019 df-rgmod 21020 df-cnfld 21234 df-refld 21468 df-dsmm 21597 df-frlm 21612 df-tng 24413 df-tcph 25017 df-rrx 25233 df-line 47577

This theorem is referenced by: (None)

W3C validator