2pthdlem1 - Metamath Proof Explorer

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Theorem 2pthdlem1 30576

Description: Lemma 1 for 2pthd 30586. (Contributed by AV, 14-Feb-2021.)

**Hypotheses**
Ref	Expression
2wlkd.p	⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩
2wlkd.f	⊢ 𝐹 = ⟨“𝐽𝐾”⟩
2wlkd.s	⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
2wlkd.n	⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶))

**Assertion**
Ref	Expression
2pthdlem1	⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^(♯‘𝐹))(𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗)))

Distinct variable groups: 𝑘,𝐹 𝑃,𝑘 𝑘,𝑉 𝑗,𝐹,𝑘 𝑃,𝑗 Allowed substitution hints: 𝜑(𝑗, 𝑘) 𝐴(𝑗, 𝑘) 𝐵(𝑗, 𝑘) 𝐶(𝑗, 𝑘) 𝐽(𝑗, 𝑘) 𝐾(𝑗, 𝑘) 𝑉(𝑗)

**Proof of Theorem 2pthdlem1**
Step	Hyp	Ref	Expression
1		2wlkd.n	. . . 4 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶))
2		2wlkd.p	. . . . 5 ⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩
3		2wlkd.f	. . . . 5 ⊢ 𝐹 = ⟨“𝐽𝐾”⟩
4		2wlkd.s	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
5	2, 3, 4	2wlkdlem3 30573	. . . 4 ⊢ (𝜑 → ((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶))
6		simpl 488	. . . . . . . . . . . 12 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) → (𝑃‘0) = 𝐴)
7		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) → (𝑃‘1) = 𝐵)
8	6, 7	neeq12d 3020	. . . . . . . . . . 11 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) → ((𝑃‘0) ≠ (𝑃‘1) ↔ 𝐴 ≠ 𝐵))
9	8	bicomd 226	. . . . . . . . . 10 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) → (𝐴 ≠ 𝐵 ↔ (𝑃‘0) ≠ (𝑃‘1)))
10	9	3adant3 1150	. . . . . . . . 9 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝐴 ≠ 𝐵 ↔ (𝑃‘0) ≠ (𝑃‘1)))
11	10	biimpcd 252	. . . . . . . 8 ⊢ (𝐴 ≠ 𝐵 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘0) ≠ (𝑃‘1)))
12	11	adantr 486	. . . . . . 7 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘0) ≠ (𝑃‘1)))
13	12	imp 412	. . . . . 6 ⊢ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶)) → (𝑃‘0) ≠ (𝑃‘1))
14	13	a1d 26	. . . . 5 ⊢ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶)) → (0 ≠ 1 → (𝑃‘0) ≠ (𝑃‘1)))
15		eqid 2763	. . . . . 6 ⊢ 1 = 1
16		eqneqall 2970	. . . . . 6 ⊢ (1 = 1 → (1 ≠ 1 → (𝑃‘1) ≠ (𝑃‘1)))
17	15, 16	mp1i 14	. . . . 5 ⊢ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶)) → (1 ≠ 1 → (𝑃‘1) ≠ (𝑃‘1)))
18		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘2) = 𝐶)
19		simpl 488	. . . . . . . . . . . 12 ⊢ (((𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘1) = 𝐵)
20	18, 19	neeq12d 3020	. . . . . . . . . . 11 ⊢ (((𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → ((𝑃‘2) ≠ (𝑃‘1) ↔ 𝐶 ≠ 𝐵))
21		necom 3012	. . . . . . . . . . 11 ⊢ (𝐶 ≠ 𝐵 ↔ 𝐵 ≠ 𝐶)
22	20, 21	bitr2di 291	. . . . . . . . . 10 ⊢ (((𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝐵 ≠ 𝐶 ↔ (𝑃‘2) ≠ (𝑃‘1)))
23	22	3adant1 1148	. . . . . . . . 9 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝐵 ≠ 𝐶 ↔ (𝑃‘2) ≠ (𝑃‘1)))
24	23	biimpcd 252	. . . . . . . 8 ⊢ (𝐵 ≠ 𝐶 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘2) ≠ (𝑃‘1)))
25	24	adantl 487	. . . . . . 7 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘2) ≠ (𝑃‘1)))
26	25	imp 412	. . . . . 6 ⊢ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶)) → (𝑃‘2) ≠ (𝑃‘1))
27	26	a1d 26	. . . . 5 ⊢ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶)) → (2 ≠ 1 → (𝑃‘2) ≠ (𝑃‘1)))
28	14, 17, 27	3jca 1146	. . . 4 ⊢ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶)) → ((0 ≠ 1 → (𝑃‘0) ≠ (𝑃‘1)) ∧ (1 ≠ 1 → (𝑃‘1) ≠ (𝑃‘1)) ∧ (2 ≠ 1 → (𝑃‘2) ≠ (𝑃‘1))))
29	1, 5, 28	syl2anc 596	. . 3 ⊢ (𝜑 → ((0 ≠ 1 → (𝑃‘0) ≠ (𝑃‘1)) ∧ (1 ≠ 1 → (𝑃‘1) ≠ (𝑃‘1)) ∧ (2 ≠ 1 → (𝑃‘2) ≠ (𝑃‘1))))
30	2	fveq2i 6919	. . . . . . . 8 ⊢ (♯‘𝑃) = (♯‘⟨“𝐴𝐵𝐶”⟩)
31		s3len 15088	. . . . . . . 8 ⊢ (♯‘⟨“𝐴𝐵𝐶”⟩) = 3
32	30, 31	eqtri 2786	. . . . . . 7 ⊢ (♯‘𝑃) = 3
33	32	oveq2i 7462	. . . . . 6 ⊢ (0..^(♯‘𝑃)) = (0..^3)
34		fzo0to3tp 13927	. . . . . 6 ⊢ (0..^3) = {0, 1, 2}
35	33, 34	eqtri 2786	. . . . 5 ⊢ (0..^(♯‘𝑃)) = {0, 1, 2}
36	35	raleqi 3321	. . . 4 ⊢ (∀𝑘 ∈ (0..^(♯‘𝑃))(𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)) ↔ ∀𝑘 ∈ {0, 1, 2} (𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)))
37		c0ex 11339	. . . . 5 ⊢ 0 ∈ V
38		1ex 11342	. . . . 5 ⊢ 1 ∈ V
39		2ex 12460	. . . . 5 ⊢ 2 ∈ V
40		neeq1 3021	. . . . . 6 ⊢ (𝑘 = 0 → (𝑘 ≠ 1 ↔ 0 ≠ 1))
41		fveq2 6916	. . . . . . 7 ⊢ (𝑘 = 0 → (𝑃‘𝑘) = (𝑃‘0))
42	41	neeq1d 3018	. . . . . 6 ⊢ (𝑘 = 0 → ((𝑃‘𝑘) ≠ (𝑃‘1) ↔ (𝑃‘0) ≠ (𝑃‘1)))
43	40, 42	imbi12d 347	. . . . 5 ⊢ (𝑘 = 0 → ((𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)) ↔ (0 ≠ 1 → (𝑃‘0) ≠ (𝑃‘1))))
44		neeq1 3021	. . . . . 6 ⊢ (𝑘 = 1 → (𝑘 ≠ 1 ↔ 1 ≠ 1))
45		fveq2 6916	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑃‘𝑘) = (𝑃‘1))
46	45	neeq1d 3018	. . . . . 6 ⊢ (𝑘 = 1 → ((𝑃‘𝑘) ≠ (𝑃‘1) ↔ (𝑃‘1) ≠ (𝑃‘1)))
47	44, 46	imbi12d 347	. . . . 5 ⊢ (𝑘 = 1 → ((𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)) ↔ (1 ≠ 1 → (𝑃‘1) ≠ (𝑃‘1))))
48		neeq1 3021	. . . . . 6 ⊢ (𝑘 = 2 → (𝑘 ≠ 1 ↔ 2 ≠ 1))
49		fveq2 6916	. . . . . . 7 ⊢ (𝑘 = 2 → (𝑃‘𝑘) = (𝑃‘2))
50	49	neeq1d 3018	. . . . . 6 ⊢ (𝑘 = 2 → ((𝑃‘𝑘) ≠ (𝑃‘1) ↔ (𝑃‘2) ≠ (𝑃‘1)))
51	48, 50	imbi12d 347	. . . . 5 ⊢ (𝑘 = 2 → ((𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)) ↔ (2 ≠ 1 → (𝑃‘2) ≠ (𝑃‘1))))
52	37, 38, 39, 43, 47, 51	raltp 4667	. . . 4 ⊢ (∀𝑘 ∈ {0, 1, 2} (𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)) ↔ ((0 ≠ 1 → (𝑃‘0) ≠ (𝑃‘1)) ∧ (1 ≠ 1 → (𝑃‘1) ≠ (𝑃‘1)) ∧ (2 ≠ 1 → (𝑃‘2) ≠ (𝑃‘1))))
53	36, 52	bitri 278	. . 3 ⊢ (∀𝑘 ∈ (0..^(♯‘𝑃))(𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)) ↔ ((0 ≠ 1 → (𝑃‘0) ≠ (𝑃‘1)) ∧ (1 ≠ 1 → (𝑃‘1) ≠ (𝑃‘1)) ∧ (2 ≠ 1 → (𝑃‘2) ≠ (𝑃‘1))))
54	29, 53	sylibr 237	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝑃))(𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)))
55	3	fveq2i 6919	. . . . . . . 8 ⊢ (♯‘𝐹) = (♯‘⟨“𝐽𝐾”⟩)
56		s2len 15082	. . . . . . . 8 ⊢ (♯‘⟨“𝐽𝐾”⟩) = 2
57	55, 56	eqtri 2786	. . . . . . 7 ⊢ (♯‘𝐹) = 2
58	57	oveq2i 7462	. . . . . 6 ⊢ (1..^(♯‘𝐹)) = (1..^2)
59		fzo12sn 13923	. . . . . 6 ⊢ (1..^2) = {1}
60	58, 59	eqtri 2786	. . . . 5 ⊢ (1..^(♯‘𝐹)) = {1}
61	60	raleqi 3321	. . . 4 ⊢ (∀𝑗 ∈ (1..^(♯‘𝐹))(𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗)) ↔ ∀𝑗 ∈ {1} (𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗)))
62		neeq2 3022	. . . . . 6 ⊢ (𝑗 = 1 → (𝑘 ≠ 𝑗 ↔ 𝑘 ≠ 1))
63		fveq2 6916	. . . . . . 7 ⊢ (𝑗 = 1 → (𝑃‘𝑗) = (𝑃‘1))
64	63	neeq2d 3019	. . . . . 6 ⊢ (𝑗 = 1 → ((𝑃‘𝑘) ≠ (𝑃‘𝑗) ↔ (𝑃‘𝑘) ≠ (𝑃‘1)))
65	62, 64	imbi12d 347	. . . . 5 ⊢ (𝑗 = 1 → ((𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗)) ↔ (𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1))))
66	38, 65	ralsn 4643	. . . 4 ⊢ (∀𝑗 ∈ {1} (𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗)) ↔ (𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)))
67	61, 66	bitri 278	. . 3 ⊢ (∀𝑗 ∈ (1..^(♯‘𝐹))(𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗)) ↔ (𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)))
68	67	ralbii 3112	. 2 ⊢ (∀𝑘 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^(♯‘𝐹))(𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗)) ↔ ∀𝑘 ∈ (0..^(♯‘𝑃))(𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)))
69	54, 68	sylibr 237	1 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^(♯‘𝐹))(𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 {csn 4588 {ctp 4592 ‘cfv 6566 (class class class)co 7451 0cc0 11239 1c1 11240 2c2 12437 3c3 12438 ..^cfzo 13828 ♯chash 14515 ⟨“cs2 15034 ⟨“cs3 15035

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-nn 12376 df-2 12445 df-3 12446 df-n0 12647 df-z 12734 df-uz 13006 df-fz 13680 df-fzo 13829 df-hash 14516 df-word 14700 df-concat 14757 df-s1 14784 df-s2 15041 df-s3 15042

This theorem is used by: 2pthd 30586

W3C validator