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Theorem 2pthdlem1 30016

Description: Lemma 1 for 2pthd 30026. (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 30013	. . . 4 ⊢ (𝜑 → ((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶))
6		simpl 483	. . . . . . . . . . . 12 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) → (𝑃‘0) = 𝐴)
7		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) → (𝑃‘1) = 𝐵)
8	6, 7	neeq12d 2995	. . . . . . . . . . 11 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) → ((𝑃‘0) ≠ (𝑃‘1) ↔ 𝐴 ≠ 𝐵))
9	8	bicomd 224	. . . . . . . . . 10 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) → (𝐴 ≠ 𝐵 ↔ (𝑃‘0) ≠ (𝑃‘1)))
10	9	3adant3 1138	. . . . . . . . 9 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝐴 ≠ 𝐵 ↔ (𝑃‘0) ≠ (𝑃‘1)))
11	10	biimpcd 250	. . . . . . . 8 ⊢ (𝐴 ≠ 𝐵 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘0) ≠ (𝑃‘1)))
12	11	adantr 481	. . . . . . 7 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘0) ≠ (𝑃‘1)))
13	12	imp 407	. . . . . 6 ⊢ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶)) → (𝑃‘0) ≠ (𝑃‘1))
14	13	a1d 25	. . . . 5 ⊢ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶)) → (0 ≠ 1 → (𝑃‘0) ≠ (𝑃‘1)))
15		eqid 2739	. . . . . 6 ⊢ 1 = 1
16		eqneqall 2945	. . . . . 6 ⊢ (1 = 1 → (1 ≠ 1 → (𝑃‘1) ≠ (𝑃‘1)))
17	15, 16	mp1i 13	. . . . 5 ⊢ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶)) → (1 ≠ 1 → (𝑃‘1) ≠ (𝑃‘1)))
18		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘2) = 𝐶)
19		simpl 483	. . . . . . . . . . . 12 ⊢ (((𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘1) = 𝐵)
20	18, 19	neeq12d 2995	. . . . . . . . . . 11 ⊢ (((𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → ((𝑃‘2) ≠ (𝑃‘1) ↔ 𝐶 ≠ 𝐵))
21		necom 2987	. . . . . . . . . . 11 ⊢ (𝐶 ≠ 𝐵 ↔ 𝐵 ≠ 𝐶)
22	20, 21	bitr2di 289	. . . . . . . . . 10 ⊢ (((𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝐵 ≠ 𝐶 ↔ (𝑃‘2) ≠ (𝑃‘1)))
23	22	3adant1 1136	. . . . . . . . 9 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝐵 ≠ 𝐶 ↔ (𝑃‘2) ≠ (𝑃‘1)))
24	23	biimpcd 250	. . . . . . . 8 ⊢ (𝐵 ≠ 𝐶 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘2) ≠ (𝑃‘1)))
25	24	adantl 482	. . . . . . 7 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘2) ≠ (𝑃‘1)))
26	25	imp 407	. . . . . 6 ⊢ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶)) → (𝑃‘2) ≠ (𝑃‘1))
27	26	a1d 25	. . . . 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 1134	. . . 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 590	. . 3 ⊢ (𝜑 → ((0 ≠ 1 → (𝑃‘0) ≠ (𝑃‘1)) ∧ (1 ≠ 1 → (𝑃‘1) ≠ (𝑃‘1)) ∧ (2 ≠ 1 → (𝑃‘2) ≠ (𝑃‘1))))
30	2	fveq2i 6830	. . . . . . . 8 ⊢ (♯‘𝑃) = (♯‘⟨“𝐴𝐵𝐶”⟩)
31		s3len 14847	. . . . . . . 8 ⊢ (♯‘⟨“𝐴𝐵𝐶”⟩) = 3
32	30, 31	eqtri 2762	. . . . . . 7 ⊢ (♯‘𝑃) = 3
33	32	oveq2i 7367	. . . . . 6 ⊢ (0..^(♯‘𝑃)) = (0..^3)
34		fzo0to3tp 13698	. . . . . 6 ⊢ (0..^3) = {0, 1, 2}
35	33, 34	eqtri 2762	. . . . 5 ⊢ (0..^(♯‘𝑃)) = {0, 1, 2}
36	35	raleqi 3295	. . . 4 ⊢ (∀𝑘 ∈ (0..^(♯‘𝑃))(𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)) ↔ ∀𝑘 ∈ {0, 1, 2} (𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)))
37		c0ex 11129	. . . . 5 ⊢ 0 ∈ V
38		1ex 11131	. . . . 5 ⊢ 1 ∈ V
39		2ex 12249	. . . . 5 ⊢ 2 ∈ V
40		neeq1 2996	. . . . . 6 ⊢ (𝑘 = 0 → (𝑘 ≠ 1 ↔ 0 ≠ 1))
41		fveq2 6827	. . . . . . 7 ⊢ (𝑘 = 0 → (𝑃‘𝑘) = (𝑃‘0))
42	41	neeq1d 2993	. . . . . 6 ⊢ (𝑘 = 0 → ((𝑃‘𝑘) ≠ (𝑃‘1) ↔ (𝑃‘0) ≠ (𝑃‘1)))
43	40, 42	imbi12d 345	. . . . 5 ⊢ (𝑘 = 0 → ((𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)) ↔ (0 ≠ 1 → (𝑃‘0) ≠ (𝑃‘1))))
44		neeq1 2996	. . . . . 6 ⊢ (𝑘 = 1 → (𝑘 ≠ 1 ↔ 1 ≠ 1))
45		fveq2 6827	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑃‘𝑘) = (𝑃‘1))
46	45	neeq1d 2993	. . . . . 6 ⊢ (𝑘 = 1 → ((𝑃‘𝑘) ≠ (𝑃‘1) ↔ (𝑃‘1) ≠ (𝑃‘1)))
47	44, 46	imbi12d 345	. . . . 5 ⊢ (𝑘 = 1 → ((𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)) ↔ (1 ≠ 1 → (𝑃‘1) ≠ (𝑃‘1))))
48		neeq1 2996	. . . . . 6 ⊢ (𝑘 = 2 → (𝑘 ≠ 1 ↔ 2 ≠ 1))
49		fveq2 6827	. . . . . . 7 ⊢ (𝑘 = 2 → (𝑃‘𝑘) = (𝑃‘2))
50	49	neeq1d 2993	. . . . . 6 ⊢ (𝑘 = 2 → ((𝑃‘𝑘) ≠ (𝑃‘1) ↔ (𝑃‘2) ≠ (𝑃‘1)))
51	48, 50	imbi12d 345	. . . . 5 ⊢ (𝑘 = 2 → ((𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)) ↔ (2 ≠ 1 → (𝑃‘2) ≠ (𝑃‘1))))
52	37, 38, 39, 43, 47, 51	raltp 4637	. . . 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 276	. . 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 235	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝑃))(𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)))
55	3	fveq2i 6830	. . . . . . . 8 ⊢ (♯‘𝐹) = (♯‘⟨“𝐽𝐾”⟩)
56		s2len 14842	. . . . . . . 8 ⊢ (♯‘⟨“𝐽𝐾”⟩) = 2
57	55, 56	eqtri 2762	. . . . . . 7 ⊢ (♯‘𝐹) = 2
58	57	oveq2i 7367	. . . . . 6 ⊢ (1..^(♯‘𝐹)) = (1..^2)
59		fzo12sn 13694	. . . . . 6 ⊢ (1..^2) = {1}
60	58, 59	eqtri 2762	. . . . 5 ⊢ (1..^(♯‘𝐹)) = {1}
61	60	raleqi 3295	. . . 4 ⊢ (∀𝑗 ∈ (1..^(♯‘𝐹))(𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗)) ↔ ∀𝑗 ∈ {1} (𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗)))
62		neeq2 2997	. . . . . 6 ⊢ (𝑗 = 1 → (𝑘 ≠ 𝑗 ↔ 𝑘 ≠ 1))
63		fveq2 6827	. . . . . . 7 ⊢ (𝑗 = 1 → (𝑃‘𝑗) = (𝑃‘1))
64	63	neeq2d 2994	. . . . . 6 ⊢ (𝑗 = 1 → ((𝑃‘𝑘) ≠ (𝑃‘𝑗) ↔ (𝑃‘𝑘) ≠ (𝑃‘1)))
65	62, 64	imbi12d 345	. . . . 5 ⊢ (𝑗 = 1 → ((𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗)) ↔ (𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1))))
66	38, 65	ralsn 4613	. . . 4 ⊢ (∀𝑗 ∈ {1} (𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗)) ↔ (𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)))
67	61, 66	bitri 276	. . 3 ⊢ (∀𝑗 ∈ (1..^(♯‘𝐹))(𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗)) ↔ (𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)))
68	67	ralbii 3085	. 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 235	1 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^(♯‘𝐹))(𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∀wral 3053 {csn 4555 {ctp 4559 ‘cfv 6485 (class class class)co 7356 0cc0 11029 1c1 11030 2c2 12227 3c3 12228 ..^cfzo 13599 ♯chash 14283 ⟨“cs2 14794 ⟨“cs3 14795

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-n0 12429 df-z 12516 df-uz 12780 df-fz 13453 df-fzo 13600 df-hash 14284 df-word 14467 df-concat 14524 df-s1 14550 df-s2 14801 df-s3 14802

This theorem is referenced by: 2pthd 30026

W3C validator