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Theorem 2pthdlem1 27391

Description: Lemma 1 for 2pthd 27401. (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 27388	. . . 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 3044	. . . . . . . . . . 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 1125	. . . . . . . . 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 2794	. . . . . 6 ⊢ 1 = 1
16		eqneqall 2994	. . . . . 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 3044	. . . . . . . . . . 11 ⊢ (((𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → ((𝑃‘2) ≠ (𝑃‘1) ↔ 𝐶 ≠ 𝐵))
21		necom 3036	. . . . . . . . . . 11 ⊢ (𝐶 ≠ 𝐵 ↔ 𝐵 ≠ 𝐶)
22	20, 21	syl6rbb 289	. . . . . . . . . 10 ⊢ (((𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝐵 ≠ 𝐶 ↔ (𝑃‘2) ≠ (𝑃‘1)))
23	22	3adant1 1123	. . . . . . . . 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 1121	. . . 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 584	. . 3 ⊢ (𝜑 → ((0 ≠ 1 → (𝑃‘0) ≠ (𝑃‘1)) ∧ (1 ≠ 1 → (𝑃‘1) ≠ (𝑃‘1)) ∧ (2 ≠ 1 → (𝑃‘2) ≠ (𝑃‘1))))
30	2	fveq2i 6544	. . . . . . . 8 ⊢ (♯‘𝑃) = (♯‘⟨“𝐴𝐵𝐶”⟩)
31		s3len 14092	. . . . . . . 8 ⊢ (♯‘⟨“𝐴𝐵𝐶”⟩) = 3
32	30, 31	eqtri 2818	. . . . . . 7 ⊢ (♯‘𝑃) = 3
33	32	oveq2i 7030	. . . . . 6 ⊢ (0..^(♯‘𝑃)) = (0..^3)
34		fzo0to3tp 12973	. . . . . 6 ⊢ (0..^3) = {0, 1, 2}
35	33, 34	eqtri 2818	. . . . 5 ⊢ (0..^(♯‘𝑃)) = {0, 1, 2}
36	35	raleqi 3372	. . . 4 ⊢ (∀𝑘 ∈ (0..^(♯‘𝑃))(𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)) ↔ ∀𝑘 ∈ {0, 1, 2} (𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)))
37		c0ex 10484	. . . . 5 ⊢ 0 ∈ V
38		1ex 10486	. . . . 5 ⊢ 1 ∈ V
39		2ex 11564	. . . . 5 ⊢ 2 ∈ V
40		neeq1 3045	. . . . . 6 ⊢ (𝑘 = 0 → (𝑘 ≠ 1 ↔ 0 ≠ 1))
41		fveq2 6541	. . . . . . 7 ⊢ (𝑘 = 0 → (𝑃‘𝑘) = (𝑃‘0))
42	41	neeq1d 3042	. . . . . 6 ⊢ (𝑘 = 0 → ((𝑃‘𝑘) ≠ (𝑃‘1) ↔ (𝑃‘0) ≠ (𝑃‘1)))
43	40, 42	imbi12d 346	. . . . 5 ⊢ (𝑘 = 0 → ((𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)) ↔ (0 ≠ 1 → (𝑃‘0) ≠ (𝑃‘1))))
44		neeq1 3045	. . . . . 6 ⊢ (𝑘 = 1 → (𝑘 ≠ 1 ↔ 1 ≠ 1))
45		fveq2 6541	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑃‘𝑘) = (𝑃‘1))
46	45	neeq1d 3042	. . . . . 6 ⊢ (𝑘 = 1 → ((𝑃‘𝑘) ≠ (𝑃‘1) ↔ (𝑃‘1) ≠ (𝑃‘1)))
47	44, 46	imbi12d 346	. . . . 5 ⊢ (𝑘 = 1 → ((𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)) ↔ (1 ≠ 1 → (𝑃‘1) ≠ (𝑃‘1))))
48		neeq1 3045	. . . . . 6 ⊢ (𝑘 = 2 → (𝑘 ≠ 1 ↔ 2 ≠ 1))
49		fveq2 6541	. . . . . . 7 ⊢ (𝑘 = 2 → (𝑃‘𝑘) = (𝑃‘2))
50	49	neeq1d 3042	. . . . . 6 ⊢ (𝑘 = 2 → ((𝑃‘𝑘) ≠ (𝑃‘1) ↔ (𝑃‘2) ≠ (𝑃‘1)))
51	48, 50	imbi12d 346	. . . . 5 ⊢ (𝑘 = 2 → ((𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)) ↔ (2 ≠ 1 → (𝑃‘2) ≠ (𝑃‘1))))
52	37, 38, 39, 43, 47, 51	raltp 4550	. . . 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 6544	. . . . . . . 8 ⊢ (♯‘𝐹) = (♯‘⟨“𝐽𝐾”⟩)
56		s2len 14087	. . . . . . . 8 ⊢ (♯‘⟨“𝐽𝐾”⟩) = 2
57	55, 56	eqtri 2818	. . . . . . 7 ⊢ (♯‘𝐹) = 2
58	57	oveq2i 7030	. . . . . 6 ⊢ (1..^(♯‘𝐹)) = (1..^2)
59		fzo12sn 12970	. . . . . 6 ⊢ (1..^2) = {1}
60	58, 59	eqtri 2818	. . . . 5 ⊢ (1..^(♯‘𝐹)) = {1}
61	60	raleqi 3372	. . . 4 ⊢ (∀𝑗 ∈ (1..^(♯‘𝐹))(𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗)) ↔ ∀𝑗 ∈ {1} (𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗)))
62		neeq2 3046	. . . . . 6 ⊢ (𝑗 = 1 → (𝑘 ≠ 𝑗 ↔ 𝑘 ≠ 1))
63		fveq2 6541	. . . . . . 7 ⊢ (𝑗 = 1 → (𝑃‘𝑗) = (𝑃‘1))
64	63	neeq2d 3043	. . . . . 6 ⊢ (𝑗 = 1 → ((𝑃‘𝑘) ≠ (𝑃‘𝑗) ↔ (𝑃‘𝑘) ≠ (𝑃‘1)))
65	62, 64	imbi12d 346	. . . . 5 ⊢ (𝑗 = 1 → ((𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗)) ↔ (𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1))))
66	38, 65	ralsn 4528	. . . 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 3131	. 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 1080 = wceq 1522 ∈ wcel 2080 ≠ wne 2983 ∀wral 3104 {csn 4474 {ctp 4478 ‘cfv 6228 (class class class)co 7019 0cc0 10386 1c1 10387 2c2 11542 3c3 11543 ..^cfzo 12883 ♯chash 13540 ⟨“cs2 14039 ⟨“cs3 14040

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-rep 5084 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-cnex 10442 ax-resscn 10443 ax-1cn 10444 ax-icn 10445 ax-addcl 10446 ax-addrcl 10447 ax-mulcl 10448 ax-mulrcl 10449 ax-mulcom 10450 ax-addass 10451 ax-mulass 10452 ax-distr 10453 ax-i2m1 10454 ax-1ne0 10455 ax-1rid 10456 ax-rnegex 10457 ax-rrecex 10458 ax-cnre 10459 ax-pre-lttri 10460 ax-pre-lttrn 10461 ax-pre-ltadd 10462 ax-pre-mulgt0 10463

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-nel 3090 df-ral 3109 df-rex 3110 df-reu 3111 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-int 4785 df-iun 4829 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-pred 6026 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-om 7440 df-1st 7548 df-2nd 7549 df-wrecs 7801 df-recs 7863 df-rdg 7901 df-1o 7956 df-oadd 7960 df-er 8142 df-en 8361 df-dom 8362 df-sdom 8363 df-fin 8364 df-card 9217 df-pnf 10526 df-mnf 10527 df-xr 10528 df-ltxr 10529 df-le 10530 df-sub 10721 df-neg 10722 df-nn 11489 df-2 11550 df-3 11551 df-n0 11748 df-z 11832 df-uz 12094 df-fz 12743 df-fzo 12884 df-hash 13541 df-word 13708 df-concat 13769 df-s1 13794 df-s2 14046 df-s3 14047

This theorem is referenced by: 2pthd 27401

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