axlowdimlem13 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > axlowdimlem13		Structured version Visualization version GIF version

Theorem axlowdimlem13 29101

Description: Lemma for axlowdim 29108. 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 12321	. . . . . . . . 9 ⊢ 2 ≠ 0
2	1	neii 2958	. . . . . . . 8 ⊢ ¬ 2 = 0
3		eqcom 2768	. . . . . . . . 9 ⊢ (2 = 0 ↔ 0 = 2)
4		1pneg1e0 12332	. . . . . . . . . . 11 ⊢ (1 + -1) = 0
5	4	eqcomi 2770	. . . . . . . . . 10 ⊢ 0 = (1 + -1)
6		df-2 12277	. . . . . . . . . 10 ⊢ 2 = (1 + 1)
7	5, 6	eqeq12i 2779	. . . . . . . . 9 ⊢ (0 = 2 ↔ (1 + -1) = (1 + 1))
8		ax-1cn 11128	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
9		neg1cn 12177	. . . . . . . . . 10 ⊢ -1 ∈ ℂ
10	8, 9, 8	addcani 11373	. . . . . . . . 9 ⊢ ((1 + -1) = (1 + 1) ↔ -1 = 1)
11	3, 7, 10	3bitri 299	. . . . . . . 8 ⊢ (2 = 0 ↔ -1 = 1)
12	2, 11	mtbi 324	. . . . . . 7 ⊢ ¬ -1 = 1
13	12	intnanr 491	. . . . . 6 ⊢ ¬ (-1 = 1 ∧ 0 = 0)
14		ax-1ne0 11139	. . . . . . . . 9 ⊢ 1 ≠ 0
15	14	neii 2958	. . . . . . . 8 ⊢ ¬ 1 = 0
16		negeq0 11482	. . . . . . . . 9 ⊢ (1 ∈ ℂ → (1 = 0 ↔ -1 = 0))
17	8, 16	ax-mp 5	. . . . . . . 8 ⊢ (1 = 0 ↔ -1 = 0)
18	15, 17	mtbi 324	. . . . . . 7 ⊢ ¬ -1 = 0
19	18	intnanr 491	. . . . . 6 ⊢ ¬ (-1 = 0 ∧ 0 = 1)
20	13, 19	pm3.2ni 891	. . . . 5 ⊢ ¬ ((-1 = 1 ∧ 0 = 0) ∨ (-1 = 0 ∧ 0 = 1))
21		negex 11425	. . . . . 6 ⊢ -1 ∈ V
22		c0ex 11170	. . . . . 6 ⊢ 0 ∈ V
23		1ex 11173	. . . . . 6 ⊢ 1 ∈ V
24	21, 22, 23, 22	preq12b 4807	. . . . 5 ⊢ ({-1, 0} = {1, 0} ↔ ((-1 = 1 ∧ 0 = 0) ∨ (-1 = 0 ∧ 0 = 1)))
25	20, 24	mtbir 325	. . . 4 ⊢ ¬ {-1, 0} = {1, 0}
26		3ex 12297	. . . . . . . . 9 ⊢ 3 ∈ V
27	26	rnsnop 6207	. . . . . . . 8 ⊢ ran {⟨3, -1⟩} = {-1}
28	27	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ran {⟨3, -1⟩} = {-1})
29		elnnuz 12876	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ_≥‘1))
30		eluzfz1 13533	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 1 ∈ (1...𝑁))
31	29, 30	sylbi 219	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 1 ∈ (1...𝑁))
32		df-3 12278	. . . . . . . . . . . . . . 15 ⊢ 3 = (2 + 1)
33		1e0p1 12732	. . . . . . . . . . . . . . 15 ⊢ 1 = (0 + 1)
34	32, 33	eqeq12i 2779	. . . . . . . . . . . . . 14 ⊢ (3 = 1 ↔ (2 + 1) = (0 + 1))
35		2cn 12290	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
36		0cn 11168	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℂ
37	35, 36, 8	addcan2i 11374	. . . . . . . . . . . . . 14 ⊢ ((2 + 1) = (0 + 1) ↔ 2 = 0)
38	34, 37	bitri 277	. . . . . . . . . . . . 13 ⊢ (3 = 1 ↔ 2 = 0)
39	38	necon3bii 3008	. . . . . . . . . . . 12 ⊢ (3 ≠ 1 ↔ 2 ≠ 0)
40	1, 39	mpbir 233	. . . . . . . . . . 11 ⊢ 3 ≠ 1
41	40	necomi 3010	. . . . . . . . . 10 ⊢ 1 ≠ 3
42		eldifsn 4745	. . . . . . . . . 10 ⊢ (1 ∈ ((1...𝑁) ∖ {3}) ↔ (1 ∈ (1...𝑁) ∧ 1 ≠ 3))
43	31, 41, 42	sylanblrc 599	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 1 ∈ ((1...𝑁) ∖ {3}))
44	43	adantr 484	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 1 ∈ ((1...𝑁) ∖ {3}))
45		ne0i 4293	. . . . . . . 8 ⊢ (1 ∈ ((1...𝑁) ∖ {3}) → ((1...𝑁) ∖ {3}) ≠ ∅)
46		rnxp 6152	. . . . . . . 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 4122	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → (ran {⟨3, -1⟩} ∪ ran (((1...𝑁) ∖ {3}) × {0})) = ({-1} ∪ {0}))
49		rnun 6126	. . . . . 6 ⊢ ran ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = (ran {⟨3, -1⟩} ∪ ran (((1...𝑁) ∖ {3}) × {0}))
50		df-pr 4584	. . . . . 6 ⊢ {-1, 0} = ({-1} ∪ {0})
51	48, 49, 50	3eqtr4g 2821	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ran ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = {-1, 0})
52		ovex 7425	. . . . . . . . 9 ⊢ (𝐼 + 1) ∈ V
53	52	rnsnop 6207	. . . . . . . 8 ⊢ ran {⟨(𝐼 + 1), 1⟩} = {1}
54	53	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ran {⟨(𝐼 + 1), 1⟩} = {1})
55		nnz 12586	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
56		fzssp1 13569	. . . . . . . . . . . 12 ⊢ (1...(𝑁 − 1)) ⊆ (1...((𝑁 − 1) + 1))
57		zcn 12570	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
58		npcan1 11609	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
59	58	oveq2d 7408	. . . . . . . . . . . . 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 3978	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → (1...(𝑁 − 1)) ⊆ (1...𝑁))
62	55, 61	syl 17	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (1...(𝑁 − 1)) ⊆ (1...𝑁))
63	62	sselda 3936	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 𝐼 ∈ (1...𝑁))
64		elfzelz 13526	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ (1...(𝑁 − 1)) → 𝐼 ∈ ℤ)
65	64	zred 12674	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (1...(𝑁 − 1)) → 𝐼 ∈ ℝ)
66		id 22	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ ℝ → 𝐼 ∈ ℝ)
67		ltp1 12028	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ ℝ → 𝐼 < (𝐼 + 1))
68	66, 67	ltned 11316	. . . . . . . . . . 11 ⊢ (𝐼 ∈ ℝ → 𝐼 ≠ (𝐼 + 1))
69	65, 68	syl 17	. . . . . . . . . 10 ⊢ (𝐼 ∈ (1...(𝑁 − 1)) → 𝐼 ≠ (𝐼 + 1))
70	69	adantl 485	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 𝐼 ≠ (𝐼 + 1))
71		eldifsn 4745	. . . . . . . . 9 ⊢ (𝐼 ∈ ((1...𝑁) ∖ {(𝐼 + 1)}) ↔ (𝐼 ∈ (1...𝑁) ∧ 𝐼 ≠ (𝐼 + 1)))
72	63, 70, 71	sylanbrc 592	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 𝐼 ∈ ((1...𝑁) ∖ {(𝐼 + 1)}))
73		ne0i 4293	. . . . . . . 8 ⊢ (𝐼 ∈ ((1...𝑁) ∖ {(𝐼 + 1)}) → ((1...𝑁) ∖ {(𝐼 + 1)}) ≠ ∅)
74		rnxp 6152	. . . . . . . 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 4122	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → (ran {⟨(𝐼 + 1), 1⟩} ∪ ran (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})) = ({1} ∪ {0}))
77		rnun 6126	. . . . . 6 ⊢ ran ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})) = (ran {⟨(𝐼 + 1), 1⟩} ∪ ran (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))
78		df-pr 4584	. . . . . 6 ⊢ {1, 0} = ({1} ∪ {0})
79	76, 77, 78	3eqtr4g 2821	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ran ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})) = {1, 0})
80	51, 79	eqeq12d 2777	. . . 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 329	. . 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 5910	. . 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 2779	. . 3 ⊢ (𝑃 = 𝑄 ↔ ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})))
87	86	necon3abii 3002	. 2 ⊢ (𝑃 ≠ 𝑄 ↔ ¬ ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})))
88	83, 87	sylibr 236	1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 𝑃 ≠ 𝑄)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∨ wo 858 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∖ cdif 3901 ∪ cun 3902 ⊆ wss 3904 ∅c0 4285 {csn 4581 {cpr 4583 ⟨cop 4587 × cxp 5643 ran crn 5646 ‘cfv 6517 (class class class)co 7392 ℂcc 11068 ℝcr 11069 0cc0 11070 1c1 11071 + caddc 11073 − cmin 11411 -cneg 11412 ℕcn 12207 2c2 12269 3c3 12270 ℤcz 12565 ℤ_≥cuz 12836 ...cfz 13509

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7843 df-1st 7966 df-2nd 7967 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-er 8673 df-en 8924 df-dom 8925 df-sdom 8926 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-nn 12208 df-2 12277 df-3 12278 df-n0 12479 df-z 12566 df-uz 12837 df-fz 13510

This theorem is referenced by: axlowdimlem15 29103

W3C validator