axlowdimlem13 - Metamath Proof Explorer

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Theorem axlowdimlem13 29285

Description: Lemma for axlowdim 29292. 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 12348	. . . . . . . . 9 ⊢ 2 ≠ 0
2	1	neii 2960	. . . . . . . 8 ⊢ ¬ 2 = 0
3		eqcom 2770	. . . . . . . . 9 ⊢ (2 = 0 ↔ 0 = 2)
4		1pneg1e0 12359	. . . . . . . . . . 11 ⊢ (1 + -1) = 0
5	4	eqcomi 2772	. . . . . . . . . 10 ⊢ 0 = (1 + -1)
6		df-2 12304	. . . . . . . . . 10 ⊢ 2 = (1 + 1)
7	5, 6	eqeq12i 2781	. . . . . . . . 9 ⊢ (0 = 2 ↔ (1 + -1) = (1 + 1))
8		ax-1cn 11159	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
9		neg1cn 12204	. . . . . . . . . 10 ⊢ -1 ∈ ℂ
10	8, 9, 8	addcani 11404	. . . . . . . . 9 ⊢ ((1 + -1) = (1 + 1) ↔ -1 = 1)
11	3, 7, 10	3bitri 300	. . . . . . . 8 ⊢ (2 = 0 ↔ -1 = 1)
12	2, 11	mtbi 325	. . . . . . 7 ⊢ ¬ -1 = 1
13	12	intnanr 492	. . . . . 6 ⊢ ¬ (-1 = 1 ∧ 0 = 0)
14		ax-1ne0 11170	. . . . . . . . 9 ⊢ 1 ≠ 0
15	14	neii 2960	. . . . . . . 8 ⊢ ¬ 1 = 0
16		negeq0 11513	. . . . . . . . 9 ⊢ (1 ∈ ℂ → (1 = 0 ↔ -1 = 0))
17	8, 16	ax-mp 5	. . . . . . . 8 ⊢ (1 = 0 ↔ -1 = 0)
18	15, 17	mtbi 325	. . . . . . 7 ⊢ ¬ -1 = 0
19	18	intnanr 492	. . . . . 6 ⊢ ¬ (-1 = 0 ∧ 0 = 1)
20	13, 19	pm3.2ni 893	. . . . 5 ⊢ ¬ ((-1 = 1 ∧ 0 = 0) ∨ (-1 = 0 ∧ 0 = 1))
21		negex 11456	. . . . . 6 ⊢ -1 ∈ V
22		c0ex 11201	. . . . . 6 ⊢ 0 ∈ V
23		1ex 11204	. . . . . 6 ⊢ 1 ∈ V
24	21, 22, 23, 22	preq12b 4816	. . . . 5 ⊢ ({-1, 0} = {1, 0} ↔ ((-1 = 1 ∧ 0 = 0) ∨ (-1 = 0 ∧ 0 = 1)))
25	20, 24	mtbir 326	. . . 4 ⊢ ¬ {-1, 0} = {1, 0}
26		3ex 12324	. . . . . . . . 9 ⊢ 3 ∈ V
27	26	rnsnop 6227	. . . . . . . 8 ⊢ ran {⟨3, -1⟩} = {-1}
28	27	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ran {⟨3, -1⟩} = {-1})
29		elnnuz 12903	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ_≥‘1))
30		eluzfz1 13560	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 1 ∈ (1...𝑁))
31	29, 30	sylbi 220	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 1 ∈ (1...𝑁))
32		df-3 12305	. . . . . . . . . . . . . . 15 ⊢ 3 = (2 + 1)
33		1e0p1 12759	. . . . . . . . . . . . . . 15 ⊢ 1 = (0 + 1)
34	32, 33	eqeq12i 2781	. . . . . . . . . . . . . 14 ⊢ (3 = 1 ↔ (2 + 1) = (0 + 1))
35		2cn 12317	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
36		0cn 11199	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℂ
37	35, 36, 8	addcan2i 11405	. . . . . . . . . . . . . 14 ⊢ ((2 + 1) = (0 + 1) ↔ 2 = 0)
38	34, 37	bitri 278	. . . . . . . . . . . . 13 ⊢ (3 = 1 ↔ 2 = 0)
39	38	necon3bii 3010	. . . . . . . . . . . 12 ⊢ (3 ≠ 1 ↔ 2 ≠ 0)
40	1, 39	mpbir 234	. . . . . . . . . . 11 ⊢ 3 ≠ 1
41	40	necomi 3012	. . . . . . . . . 10 ⊢ 1 ≠ 3
42		eldifsn 4754	. . . . . . . . . 10 ⊢ (1 ∈ ((1...𝑁) ∖ {3}) ↔ (1 ∈ (1...𝑁) ∧ 1 ≠ 3))
43	31, 41, 42	sylanblrc 601	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 1 ∈ ((1...𝑁) ∖ {3}))
44	43	adantr 485	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 1 ∈ ((1...𝑁) ∖ {3}))
45		ne0i 4295	. . . . . . . 8 ⊢ (1 ∈ ((1...𝑁) ∖ {3}) → ((1...𝑁) ∖ {3}) ≠ ∅)
46		rnxp 6170	. . . . . . . 8 ⊢ (((1...𝑁) ∖ {3}) ≠ ∅ → ran (((1...𝑁) ∖ {3}) × {0}) = {0})
47	44, 45, 46	3syl 19	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ran (((1...𝑁) ∖ {3}) × {0}) = {0})
48	28, 47	uneq12d 4124	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → (ran {⟨3, -1⟩} ∪ ran (((1...𝑁) ∖ {3}) × {0})) = ({-1} ∪ {0}))
49		rnun 6144	. . . . . 6 ⊢ ran ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = (ran {⟨3, -1⟩} ∪ ran (((1...𝑁) ∖ {3}) × {0}))
50		df-pr 4593	. . . . . 6 ⊢ {-1, 0} = ({-1} ∪ {0})
51	48, 49, 50	3eqtr4g 2823	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ran ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = {-1, 0})
52		ovex 7445	. . . . . . . . 9 ⊢ (𝐼 + 1) ∈ V
53	52	rnsnop 6227	. . . . . . . 8 ⊢ ran {⟨(𝐼 + 1), 1⟩} = {1}
54	53	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ran {⟨(𝐼 + 1), 1⟩} = {1})
55		nnz 12613	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
56		fzssp1 13597	. . . . . . . . . . . 12 ⊢ (1...(𝑁 − 1)) ⊆ (1...((𝑁 − 1) + 1))
57		zcn 12597	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
58		npcan1 11640	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
59	58	oveq2d 7428	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℂ → (1...((𝑁 − 1) + 1)) = (1...𝑁))
60	57, 59	syl 18	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℤ → (1...((𝑁 − 1) + 1)) = (1...𝑁))
61	56, 60	sseqtrid 3980	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → (1...(𝑁 − 1)) ⊆ (1...𝑁))
62	55, 61	syl 18	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (1...(𝑁 − 1)) ⊆ (1...𝑁))
63	62	sselda 3938	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 𝐼 ∈ (1...𝑁))
64		elfzelz 13553	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ (1...(𝑁 − 1)) → 𝐼 ∈ ℤ)
65	64	zred 12701	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (1...(𝑁 − 1)) → 𝐼 ∈ ℝ)
66		id 23	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ ℝ → 𝐼 ∈ ℝ)
67		ltp1 12056	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ ℝ → 𝐼 < (𝐼 + 1))
68	66, 67	ltned 11347	. . . . . . . . . . 11 ⊢ (𝐼 ∈ ℝ → 𝐼 ≠ (𝐼 + 1))
69	65, 68	syl 18	. . . . . . . . . 10 ⊢ (𝐼 ∈ (1...(𝑁 − 1)) → 𝐼 ≠ (𝐼 + 1))
70	69	adantl 486	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 𝐼 ≠ (𝐼 + 1))
71		eldifsn 4754	. . . . . . . . 9 ⊢ (𝐼 ∈ ((1...𝑁) ∖ {(𝐼 + 1)}) ↔ (𝐼 ∈ (1...𝑁) ∧ 𝐼 ≠ (𝐼 + 1)))
72	63, 70, 71	sylanbrc 594	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 𝐼 ∈ ((1...𝑁) ∖ {(𝐼 + 1)}))
73		ne0i 4295	. . . . . . . 8 ⊢ (𝐼 ∈ ((1...𝑁) ∖ {(𝐼 + 1)}) → ((1...𝑁) ∖ {(𝐼 + 1)}) ≠ ∅)
74		rnxp 6170	. . . . . . . 8 ⊢ (((1...𝑁) ∖ {(𝐼 + 1)}) ≠ ∅ → ran (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}) = {0})
75	72, 73, 74	3syl 19	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ran (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}) = {0})
76	54, 75	uneq12d 4124	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → (ran {⟨(𝐼 + 1), 1⟩} ∪ ran (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})) = ({1} ∪ {0}))
77		rnun 6144	. . . . . 6 ⊢ ran ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})) = (ran {⟨(𝐼 + 1), 1⟩} ∪ ran (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}))
78		df-pr 4593	. . . . . 6 ⊢ {1, 0} = ({1} ∪ {0})
79	76, 77, 78	3eqtr4g 2823	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ran ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})) = {1, 0})
80	51, 79	eqeq12d 2779	. . . 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 330	. . 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 5928	. . 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 141	. 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 2781	. . 3 ⊢ (𝑃 = 𝑄 ↔ ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})))
87	86	necon3abii 3004	. 2 ⊢ (𝑃 ≠ 𝑄 ↔ ¬ ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0})) = ({⟨(𝐼 + 1), 1⟩} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})))
88	83, 87	sylibr 237	1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 𝑃 ≠ 𝑄)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∖ cdif 3903 ∪ cun 3904 ⊆ wss 3906 ∅c0 4287 {csn 4590 {cpr 4592 ⟨cop 4596 × cxp 5661 ran crn 5664 ‘cfv 6538 (class class class)co 7412 ℂcc 11099 ℝcr 11100 0cc0 11101 1c1 11102 + caddc 11104 − cmin 11442 -cneg 11443 ℕcn 12234 2c2 12296 3c3 12297 ℤcz 12592 ℤ_≥cuz 12863 ...cfz 13536

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-er 8695 df-en 8945 df-dom 8946 df-sdom 8947 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-nn 12235 df-2 12304 df-3 12305 df-n0 12506 df-z 12593 df-uz 12864 df-fz 13537

This theorem is referenced by: axlowdimlem15 29287

W3C validator