axlowdimlem13 - Metamath Proof Explorer

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Theorem axlowdimlem13 28888

Description: Lemma for axlowdim 28895. 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 12297	. . . . . . . . 9 ⊢ 2 ≠ 0
2	1	neii 2928	. . . . . . . 8 ⊢ ¬ 2 = 0
3		eqcom 2737	. . . . . . . . 9 ⊢ (2 = 0 ↔ 0 = 2)
4		1pneg1e0 12307	. . . . . . . . . . 11 ⊢ (1 + -1) = 0
5	4	eqcomi 2739	. . . . . . . . . 10 ⊢ 0 = (1 + -1)
6		df-2 12256	. . . . . . . . . 10 ⊢ 2 = (1 + 1)
7	5, 6	eqeq12i 2748	. . . . . . . . 9 ⊢ (0 = 2 ↔ (1 + -1) = (1 + 1))
8		ax-1cn 11133	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
9		neg1cn 12178	. . . . . . . . . 10 ⊢ -1 ∈ ℂ
10	8, 9, 8	addcani 11374	. . . . . . . . 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 11144	. . . . . . . . 9 ⊢ 1 ≠ 0
15	14	neii 2928	. . . . . . . 8 ⊢ ¬ 1 = 0
16		negeq0 11483	. . . . . . . . 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 11426	. . . . . 6 ⊢ -1 ∈ V
22		c0ex 11175	. . . . . 6 ⊢ 0 ∈ V
23		1ex 11177	. . . . . 6 ⊢ 1 ∈ V
24	21, 22, 23, 22	preq12b 4817	. . . . 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 12275	. . . . . . . . 9 ⊢ 3 ∈ V
27	26	rnsnop 6200	. . . . . . . 8 ⊢ ran {⟨3, -1⟩} = {-1}
28	27	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ran {⟨3, -1⟩} = {-1})
29		elnnuz 12844	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ_≥‘1))
30		eluzfz1 13499	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 1 ∈ (1...𝑁))
31	29, 30	sylbi 217	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 1 ∈ (1...𝑁))
32		df-3 12257	. . . . . . . . . . . . . . 15 ⊢ 3 = (2 + 1)
33		1e0p1 12698	. . . . . . . . . . . . . . 15 ⊢ 1 = (0 + 1)
34	32, 33	eqeq12i 2748	. . . . . . . . . . . . . 14 ⊢ (3 = 1 ↔ (2 + 1) = (0 + 1))
35		2cn 12268	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
36		0cn 11173	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℂ
37	35, 36, 8	addcan2i 11375	. . . . . . . . . . . . . 14 ⊢ ((2 + 1) = (0 + 1) ↔ 2 = 0)
38	34, 37	bitri 275	. . . . . . . . . . . . 13 ⊢ (3 = 1 ↔ 2 = 0)
39	38	necon3bii 2978	. . . . . . . . . . . 12 ⊢ (3 ≠ 1 ↔ 2 ≠ 0)
40	1, 39	mpbir 231	. . . . . . . . . . 11 ⊢ 3 ≠ 1
41	40	necomi 2980	. . . . . . . . . 10 ⊢ 1 ≠ 3
42		eldifsn 4753	. . . . . . . . . 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 4307	. . . . . . . 8 ⊢ (1 ∈ ((1...𝑁) ∖ {3}) → ((1...𝑁) ∖ {3}) ≠ ∅)
46		rnxp 6146	. . . . . . . 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 4135	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → (ran {⟨3, -1⟩} ∪ ran (((1...𝑁) ∖ {3}) × {0})) = ({-1} ∪ {0}))
49		rnun 6121	. . . . . 6 ⊢ ran ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = (ran {⟨3, -1⟩} ∪ ran (((1...𝑁) ∖ {3}) × {0}))
50		df-pr 4595	. . . . . 6 ⊢ {-1, 0} = ({-1} ∪ {0})
51	48, 49, 50	3eqtr4g 2790	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ran ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = {-1, 0})
52		ovex 7423	. . . . . . . . 9 ⊢ (𝐼 + 1) ∈ V
53	52	rnsnop 6200	. . . . . . . 8 ⊢ ran {⟨(𝐼 + 1), 1⟩} = {1}
54	53	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ran {⟨(𝐼 + 1), 1⟩} = {1})
55		nnz 12557	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
56		fzssp1 13535	. . . . . . . . . . . 12 ⊢ (1...(𝑁 − 1)) ⊆ (1...((𝑁 − 1) + 1))
57		zcn 12541	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
58		npcan1 11610	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
59	58	oveq2d 7406	. . . . . . . . . . . . 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 3992	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → (1...(𝑁 − 1)) ⊆ (1...𝑁))
62	55, 61	syl 17	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (1...(𝑁 − 1)) ⊆ (1...𝑁))
63	62	sselda 3949	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 𝐼 ∈ (1...𝑁))
64		elfzelz 13492	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ (1...(𝑁 − 1)) → 𝐼 ∈ ℤ)
65	64	zred 12645	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (1...(𝑁 − 1)) → 𝐼 ∈ ℝ)
66		id 22	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ ℝ → 𝐼 ∈ ℝ)
67		ltp1 12029	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ ℝ → 𝐼 < (𝐼 + 1))
68	66, 67	ltned 11317	. . . . . . . . . . 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 4753	. . . . . . . . 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 4307	. . . . . . . 8 ⊢ (𝐼 ∈ ((1...𝑁) ∖ {(𝐼 + 1)}) → ((1...𝑁) ∖ {(𝐼 + 1)}) ≠ ∅)
74		rnxp 6146	. . . . . . . 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 4135	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → (ran {⟨(𝐼 + 1), 1⟩} ∪ ran (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})) = ({1} ∪ {0}))
77		rnun 6121	. . . . . 6 ⊢ ran ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})) = (ran {⟨(𝐼 + 1), 1⟩} ∪ ran (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))
78		df-pr 4595	. . . . . 6 ⊢ {1, 0} = ({1} ∪ {0})
79	76, 77, 78	3eqtr4g 2790	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ran ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})) = {1, 0})
80	51, 79	eqeq12d 2746	. . . 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 5903	. . 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 2748	. . 3 ⊢ (𝑃 = 𝑄 ↔ ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})))
87	86	necon3abii 2972	. 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 2926 ∖ cdif 3914 ∪ cun 3915 ⊆ wss 3917 ∅c0 4299 {csn 4592 {cpr 4594 ⟨cop 4598 × cxp 5639 ran crn 5642 ‘cfv 6514 (class class class)co 7390 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 − cmin 11412 -cneg 11413 ℕcn 12193 2c2 12248 3c3 12249 ℤcz 12536 ℤ_≥cuz 12800 ...cfz 13475

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 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-n0 12450 df-z 12537 df-uz 12801 df-fz 13476

This theorem is referenced by: axlowdimlem15 28890

W3C validator