axlowdimlem13 - Metamath Proof Explorer

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Theorem axlowdimlem13 28881

Description: Lemma for axlowdim 28888. Establish that 𝑃 and 𝑄 are different points. (Contributed by Scott Fenton, 21-Apr-2013.)

**Hypotheses**
Ref	Expression
axlowdimlem13.1	⊢ 𝑃 = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))
axlowdimlem13.2	⊢ 𝑄 = ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))

**Assertion**
Ref	Expression
axlowdimlem13	⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 𝑃 ≠ 𝑄)

**Proof of Theorem axlowdimlem13**
Step	Hyp	Ref	Expression
1		2ne0 12290	. . . . . . . . 9 ⊢ 2 ≠ 0
2	1	neii 2927	. . . . . . . 8 ⊢ ¬ 2 = 0
3		eqcom 2736	. . . . . . . . 9 ⊢ (2 = 0 ↔ 0 = 2)
4		1pneg1e0 12300	. . . . . . . . . . 11 ⊢ (1 + -1) = 0
5	4	eqcomi 2738	. . . . . . . . . 10 ⊢ 0 = (1 + -1)
6		df-2 12249	. . . . . . . . . 10 ⊢ 2 = (1 + 1)
7	5, 6	eqeq12i 2747	. . . . . . . . 9 ⊢ (0 = 2 ↔ (1 + -1) = (1 + 1))
8		ax-1cn 11126	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
9		neg1cn 12171	. . . . . . . . . 10 ⊢ -1 ∈ ℂ
10	8, 9, 8	addcani 11367	. . . . . . . . 9 ⊢ ((1 + -1) = (1 + 1) ↔ -1 = 1)
11	3, 7, 10	3bitri 297	. . . . . . . 8 ⊢ (2 = 0 ↔ -1 = 1)
12	2, 11	mtbi 322	. . . . . . 7 ⊢ ¬ -1 = 1
13	12	intnanr 487	. . . . . 6 ⊢ ¬ (-1 = 1 ∧ 0 = 0)
14		ax-1ne0 11137	. . . . . . . . 9 ⊢ 1 ≠ 0
15	14	neii 2927	. . . . . . . 8 ⊢ ¬ 1 = 0
16		negeq0 11476	. . . . . . . . 9 ⊢ (1 ∈ ℂ → (1 = 0 ↔ -1 = 0))
17	8, 16	ax-mp 5	. . . . . . . 8 ⊢ (1 = 0 ↔ -1 = 0)
18	15, 17	mtbi 322	. . . . . . 7 ⊢ ¬ -1 = 0
19	18	intnanr 487	. . . . . 6 ⊢ ¬ (-1 = 0 ∧ 0 = 1)
20	13, 19	pm3.2ni 880	. . . . 5 ⊢ ¬ ((-1 = 1 ∧ 0 = 0) ∨ (-1 = 0 ∧ 0 = 1))
21		negex 11419	. . . . . 6 ⊢ -1 ∈ V
22		c0ex 11168	. . . . . 6 ⊢ 0 ∈ V
23		1ex 11170	. . . . . 6 ⊢ 1 ∈ V
24	21, 22, 23, 22	preq12b 4814	. . . . 5 ⊢ ({-1, 0} = {1, 0} ↔ ((-1 = 1 ∧ 0 = 0) ∨ (-1 = 0 ∧ 0 = 1)))
25	20, 24	mtbir 323	. . . 4 ⊢ ¬ {-1, 0} = {1, 0}
26		3ex 12268	. . . . . . . . 9 ⊢ 3 ∈ V
27	26	rnsnop 6197	. . . . . . . 8 ⊢ ran {⟨3, -1⟩} = {-1}
28	27	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ran {⟨3, -1⟩} = {-1})
29		elnnuz 12837	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ_≥‘1))
30		eluzfz1 13492	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 1 ∈ (1...𝑁))
31	29, 30	sylbi 217	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 1 ∈ (1...𝑁))
32		df-3 12250	. . . . . . . . . . . . . . 15 ⊢ 3 = (2 + 1)
33		1e0p1 12691	. . . . . . . . . . . . . . 15 ⊢ 1 = (0 + 1)
34	32, 33	eqeq12i 2747	. . . . . . . . . . . . . 14 ⊢ (3 = 1 ↔ (2 + 1) = (0 + 1))
35		2cn 12261	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
36		0cn 11166	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℂ
37	35, 36, 8	addcan2i 11368	. . . . . . . . . . . . . 14 ⊢ ((2 + 1) = (0 + 1) ↔ 2 = 0)
38	34, 37	bitri 275	. . . . . . . . . . . . 13 ⊢ (3 = 1 ↔ 2 = 0)
39	38	necon3bii 2977	. . . . . . . . . . . 12 ⊢ (3 ≠ 1 ↔ 2 ≠ 0)
40	1, 39	mpbir 231	. . . . . . . . . . 11 ⊢ 3 ≠ 1
41	40	necomi 2979	. . . . . . . . . 10 ⊢ 1 ≠ 3
42		eldifsn 4750	. . . . . . . . . 10 ⊢ (1 ∈ ((1...𝑁) ∖ {3}) ↔ (1 ∈ (1...𝑁) ∧ 1 ≠ 3))
43	31, 41, 42	sylanblrc 590	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 1 ∈ ((1...𝑁) ∖ {3}))
44	43	adantr 480	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 1 ∈ ((1...𝑁) ∖ {3}))
45		ne0i 4304	. . . . . . . 8 ⊢ (1 ∈ ((1...𝑁) ∖ {3}) → ((1...𝑁) ∖ {3}) ≠ ∅)
46		rnxp 6143	. . . . . . . 8 ⊢ (((1...𝑁) ∖ {3}) ≠ ∅ → ran (((1...𝑁) ∖ {3}) × {0}) = {0})
47	44, 45, 46	3syl 18	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ran (((1...𝑁) ∖ {3}) × {0}) = {0})
48	28, 47	uneq12d 4132	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → (ran {⟨3, -1⟩} ∪ ran (((1...𝑁) ∖ {3}) × {0})) = ({-1} ∪ {0}))
49		rnun 6118	. . . . . 6 ⊢ ran ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = (ran {⟨3, -1⟩} ∪ ran (((1...𝑁) ∖ {3}) × {0}))
50		df-pr 4592	. . . . . 6 ⊢ {-1, 0} = ({-1} ∪ {0})
51	48, 49, 50	3eqtr4g 2789	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ran ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = {-1, 0})
52		ovex 7420	. . . . . . . . 9 ⊢ (𝐼 + 1) ∈ V
53	52	rnsnop 6197	. . . . . . . 8 ⊢ ran {⟨(𝐼 + 1), 1⟩} = {1}
54	53	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ran {⟨(𝐼 + 1), 1⟩} = {1})
55		nnz 12550	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
56		fzssp1 13528	. . . . . . . . . . . 12 ⊢ (1...(𝑁 − 1)) ⊆ (1...((𝑁 − 1) + 1))
57		zcn 12534	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
58		npcan1 11603	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
59	58	oveq2d 7403	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℂ → (1...((𝑁 − 1) + 1)) = (1...𝑁))
60	57, 59	syl 17	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℤ → (1...((𝑁 − 1) + 1)) = (1...𝑁))
61	56, 60	sseqtrid 3989	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → (1...(𝑁 − 1)) ⊆ (1...𝑁))
62	55, 61	syl 17	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (1...(𝑁 − 1)) ⊆ (1...𝑁))
63	62	sselda 3946	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 𝐼 ∈ (1...𝑁))
64		elfzelz 13485	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ (1...(𝑁 − 1)) → 𝐼 ∈ ℤ)
65	64	zred 12638	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (1...(𝑁 − 1)) → 𝐼 ∈ ℝ)
66		id 22	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ ℝ → 𝐼 ∈ ℝ)
67		ltp1 12022	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ ℝ → 𝐼 < (𝐼 + 1))
68	66, 67	ltned 11310	. . . . . . . . . . 11 ⊢ (𝐼 ∈ ℝ → 𝐼 ≠ (𝐼 + 1))
69	65, 68	syl 17	. . . . . . . . . 10 ⊢ (𝐼 ∈ (1...(𝑁 − 1)) → 𝐼 ≠ (𝐼 + 1))
70	69	adantl 481	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 𝐼 ≠ (𝐼 + 1))
71		eldifsn 4750	. . . . . . . . 9 ⊢ (𝐼 ∈ ((1...𝑁) ∖ {(𝐼 + 1)}) ↔ (𝐼 ∈ (1...𝑁) ∧ 𝐼 ≠ (𝐼 + 1)))
72	63, 70, 71	sylanbrc 583	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 𝐼 ∈ ((1...𝑁) ∖ {(𝐼 + 1)}))
73		ne0i 4304	. . . . . . . 8 ⊢ (𝐼 ∈ ((1...𝑁) ∖ {(𝐼 + 1)}) → ((1...𝑁) ∖ {(𝐼 + 1)}) ≠ ∅)
74		rnxp 6143	. . . . . . . 8 ⊢ (((1...𝑁) ∖ {(𝐼 + 1)}) ≠ ∅ → ran (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}) = {0})
75	72, 73, 74	3syl 18	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ran (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}) = {0})
76	54, 75	uneq12d 4132	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → (ran {⟨(𝐼 + 1), 1⟩} ∪ ran (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})) = ({1} ∪ {0}))
77		rnun 6118	. . . . . 6 ⊢ ran ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})) = (ran {⟨(𝐼 + 1), 1⟩} ∪ ran (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))
78		df-pr 4592	. . . . . 6 ⊢ {1, 0} = ({1} ∪ {0})
79	76, 77, 78	3eqtr4g 2789	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ran ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})) = {1, 0})
80	51, 79	eqeq12d 2745	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → (ran ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ran ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})) ↔ {-1, 0} = {1, 0}))
81	25, 80	mtbiri 327	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ¬ ran ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ran ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})))
82		rneq 5900	. . 3 ⊢ (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})) → ran ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ran ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})))
83	81, 82	nsyl 140	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ¬ ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})))
84		axlowdimlem13.1	. . . 4 ⊢ 𝑃 = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))
85		axlowdimlem13.2	. . . 4 ⊢ 𝑄 = ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))
86	84, 85	eqeq12i 2747	. . 3 ⊢ (𝑃 = 𝑄 ↔ ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})))
87	86	necon3abii 2971	. 2 ⊢ (𝑃 ≠ 𝑄 ↔ ¬ ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})))
88	83, 87	sylibr 234	1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 𝑃 ≠ 𝑄)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∖ cdif 3911 ∪ cun 3912 ⊆ wss 3914 ∅c0 4296 {csn 4589 {cpr 4591 ⟨cop 4595 × cxp 5636 ran crn 5639 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 − cmin 11405 -cneg 11406 ℕcn 12186 2c2 12241 3c3 12242 ℤcz 12529 ℤ_≥cuz 12793 ...cfz 13468

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-z 12530 df-uz 12794 df-fz 13469

This theorem is referenced by: axlowdimlem15 28883

W3C validator