axlowdimlem13 - Metamath Proof Explorer

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Theorem axlowdimlem13 29023

Description: Lemma for axlowdim 29030. 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 12285	. . . . . . . . 9 ⊢ 2 ≠ 0
2	1	neii 2934	. . . . . . . 8 ⊢ ¬ 2 = 0
3		eqcom 2743	. . . . . . . . 9 ⊢ (2 = 0 ↔ 0 = 2)
4		1pneg1e0 12295	. . . . . . . . . . 11 ⊢ (1 + -1) = 0
5	4	eqcomi 2745	. . . . . . . . . 10 ⊢ 0 = (1 + -1)
6		df-2 12244	. . . . . . . . . 10 ⊢ 2 = (1 + 1)
7	5, 6	eqeq12i 2754	. . . . . . . . 9 ⊢ (0 = 2 ↔ (1 + -1) = (1 + 1))
8		ax-1cn 11096	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
9		neg1cn 12144	. . . . . . . . . 10 ⊢ -1 ∈ ℂ
10	8, 9, 8	addcani 11339	. . . . . . . . 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 11107	. . . . . . . . 9 ⊢ 1 ≠ 0
15	14	neii 2934	. . . . . . . 8 ⊢ ¬ 1 = 0
16		negeq0 11448	. . . . . . . . 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 881	. . . . 5 ⊢ ¬ ((-1 = 1 ∧ 0 = 0) ∨ (-1 = 0 ∧ 0 = 1))
21		negex 11391	. . . . . 6 ⊢ -1 ∈ V
22		c0ex 11138	. . . . . 6 ⊢ 0 ∈ V
23		1ex 11140	. . . . . 6 ⊢ 1 ∈ V
24	21, 22, 23, 22	preq12b 4793	. . . . 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 12263	. . . . . . . . 9 ⊢ 3 ∈ V
27	26	rnsnop 6188	. . . . . . . 8 ⊢ ran {⟨3, -1⟩} = {-1}
28	27	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ran {⟨3, -1⟩} = {-1})
29		elnnuz 12828	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ_≥‘1))
30		eluzfz1 13485	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 1 ∈ (1...𝑁))
31	29, 30	sylbi 217	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 1 ∈ (1...𝑁))
32		df-3 12245	. . . . . . . . . . . . . . 15 ⊢ 3 = (2 + 1)
33		1e0p1 12686	. . . . . . . . . . . . . . 15 ⊢ 1 = (0 + 1)
34	32, 33	eqeq12i 2754	. . . . . . . . . . . . . 14 ⊢ (3 = 1 ↔ (2 + 1) = (0 + 1))
35		2cn 12256	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
36		0cn 11136	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℂ
37	35, 36, 8	addcan2i 11340	. . . . . . . . . . . . . 14 ⊢ ((2 + 1) = (0 + 1) ↔ 2 = 0)
38	34, 37	bitri 275	. . . . . . . . . . . . 13 ⊢ (3 = 1 ↔ 2 = 0)
39	38	necon3bii 2984	. . . . . . . . . . . 12 ⊢ (3 ≠ 1 ↔ 2 ≠ 0)
40	1, 39	mpbir 231	. . . . . . . . . . 11 ⊢ 3 ≠ 1
41	40	necomi 2986	. . . . . . . . . 10 ⊢ 1 ≠ 3
42		eldifsn 4731	. . . . . . . . . 10 ⊢ (1 ∈ ((1...𝑁) ∖ {3}) ↔ (1 ∈ (1...𝑁) ∧ 1 ≠ 3))
43	31, 41, 42	sylanblrc 591	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 1 ∈ ((1...𝑁) ∖ {3}))
44	43	adantr 480	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 1 ∈ ((1...𝑁) ∖ {3}))
45		ne0i 4281	. . . . . . . 8 ⊢ (1 ∈ ((1...𝑁) ∖ {3}) → ((1...𝑁) ∖ {3}) ≠ ∅)
46		rnxp 6134	. . . . . . . 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 4109	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → (ran {⟨3, -1⟩} ∪ ran (((1...𝑁) ∖ {3}) × {0})) = ({-1} ∪ {0}))
49		rnun 6109	. . . . . 6 ⊢ ran ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = (ran {⟨3, -1⟩} ∪ ran (((1...𝑁) ∖ {3}) × {0}))
50		df-pr 4570	. . . . . 6 ⊢ {-1, 0} = ({-1} ∪ {0})
51	48, 49, 50	3eqtr4g 2796	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ran ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = {-1, 0})
52		ovex 7400	. . . . . . . . 9 ⊢ (𝐼 + 1) ∈ V
53	52	rnsnop 6188	. . . . . . . 8 ⊢ ran {⟨(𝐼 + 1), 1⟩} = {1}
54	53	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ran {⟨(𝐼 + 1), 1⟩} = {1})
55		nnz 12545	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
56		fzssp1 13521	. . . . . . . . . . . 12 ⊢ (1...(𝑁 − 1)) ⊆ (1...((𝑁 − 1) + 1))
57		zcn 12529	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
58		npcan1 11575	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
59	58	oveq2d 7383	. . . . . . . . . . . . 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 3964	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → (1...(𝑁 − 1)) ⊆ (1...𝑁))
62	55, 61	syl 17	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (1...(𝑁 − 1)) ⊆ (1...𝑁))
63	62	sselda 3921	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 𝐼 ∈ (1...𝑁))
64		elfzelz 13478	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ (1...(𝑁 − 1)) → 𝐼 ∈ ℤ)
65	64	zred 12633	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (1...(𝑁 − 1)) → 𝐼 ∈ ℝ)
66		id 22	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ ℝ → 𝐼 ∈ ℝ)
67		ltp1 11995	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ ℝ → 𝐼 < (𝐼 + 1))
68	66, 67	ltned 11282	. . . . . . . . . . 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 4731	. . . . . . . . 9 ⊢ (𝐼 ∈ ((1...𝑁) ∖ {(𝐼 + 1)}) ↔ (𝐼 ∈ (1...𝑁) ∧ 𝐼 ≠ (𝐼 + 1)))
72	63, 70, 71	sylanbrc 584	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 𝐼 ∈ ((1...𝑁) ∖ {(𝐼 + 1)}))
73		ne0i 4281	. . . . . . . 8 ⊢ (𝐼 ∈ ((1...𝑁) ∖ {(𝐼 + 1)}) → ((1...𝑁) ∖ {(𝐼 + 1)}) ≠ ∅)
74		rnxp 6134	. . . . . . . 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 4109	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → (ran {⟨(𝐼 + 1), 1⟩} ∪ ran (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})) = ({1} ∪ {0}))
77		rnun 6109	. . . . . 6 ⊢ ran ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})) = (ran {⟨(𝐼 + 1), 1⟩} ∪ ran (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))
78		df-pr 4570	. . . . . 6 ⊢ {1, 0} = ({1} ∪ {0})
79	76, 77, 78	3eqtr4g 2796	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ran ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})) = {1, 0})
80	51, 79	eqeq12d 2752	. . . 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 5891	. . 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 2754	. . 3 ⊢ (𝑃 = 𝑄 ↔ ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})))
87	86	necon3abii 2978	. 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 848 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∖ cdif 3886 ∪ cun 3887 ⊆ wss 3889 ∅c0 4273 {csn 4567 {cpr 4569 ⟨cop 4573 × cxp 5629 ran crn 5632 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 − cmin 11377 -cneg 11378 ℕcn 12174 2c2 12236 3c3 12237 ℤcz 12524 ℤ_≥cuz 12788 ...cfz 13461

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-z 12525 df-uz 12789 df-fz 13462

This theorem is referenced by: axlowdimlem15 29025

W3C validator