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Theorem 2pthdlem1 29998

Description: Lemma 1 for 2pthd 30008. (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 29995	. . . 4 ⊢ (𝜑 → ((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶))
6		simpl 482	. . . . . . . . . . . 12 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) → (𝑃‘0) = 𝐴)
7		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) → (𝑃‘1) = 𝐵)
8	6, 7	neeq12d 2993	. . . . . . . . . . 11 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) → ((𝑃‘0) ≠ (𝑃‘1) ↔ 𝐴 ≠ 𝐵))
9	8	bicomd 223	. . . . . . . . . 10 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) → (𝐴 ≠ 𝐵 ↔ (𝑃‘0) ≠ (𝑃‘1)))
10	9	3adant3 1133	. . . . . . . . 9 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝐴 ≠ 𝐵 ↔ (𝑃‘0) ≠ (𝑃‘1)))
11	10	biimpcd 249	. . . . . . . 8 ⊢ (𝐴 ≠ 𝐵 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘0) ≠ (𝑃‘1)))
12	11	adantr 480	. . . . . . 7 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘0) ≠ (𝑃‘1)))
13	12	imp 406	. . . . . 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 2736	. . . . . 6 ⊢ 1 = 1
16		eqneqall 2943	. . . . . 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 484	. . . . . . . . . . . 12 ⊢ (((𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘2) = 𝐶)
19		simpl 482	. . . . . . . . . . . 12 ⊢ (((𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘1) = 𝐵)
20	18, 19	neeq12d 2993	. . . . . . . . . . 11 ⊢ (((𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → ((𝑃‘2) ≠ (𝑃‘1) ↔ 𝐶 ≠ 𝐵))
21		necom 2985	. . . . . . . . . . 11 ⊢ (𝐶 ≠ 𝐵 ↔ 𝐵 ≠ 𝐶)
22	20, 21	bitr2di 288	. . . . . . . . . 10 ⊢ (((𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝐵 ≠ 𝐶 ↔ (𝑃‘2) ≠ (𝑃‘1)))
23	22	3adant1 1131	. . . . . . . . 9 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝐵 ≠ 𝐶 ↔ (𝑃‘2) ≠ (𝑃‘1)))
24	23	biimpcd 249	. . . . . . . 8 ⊢ (𝐵 ≠ 𝐶 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘2) ≠ (𝑃‘1)))
25	24	adantl 481	. . . . . . 7 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘2) ≠ (𝑃‘1)))
26	25	imp 406	. . . . . 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 1129	. . . 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 585	. . 3 ⊢ (𝜑 → ((0 ≠ 1 → (𝑃‘0) ≠ (𝑃‘1)) ∧ (1 ≠ 1 → (𝑃‘1) ≠ (𝑃‘1)) ∧ (2 ≠ 1 → (𝑃‘2) ≠ (𝑃‘1))))
30	2	fveq2i 6843	. . . . . . . 8 ⊢ (♯‘𝑃) = (♯‘⟨“𝐴𝐵𝐶”⟩)
31		s3len 14856	. . . . . . . 8 ⊢ (♯‘⟨“𝐴𝐵𝐶”⟩) = 3
32	30, 31	eqtri 2759	. . . . . . 7 ⊢ (♯‘𝑃) = 3
33	32	oveq2i 7378	. . . . . 6 ⊢ (0..^(♯‘𝑃)) = (0..^3)
34		fzo0to3tp 13707	. . . . . 6 ⊢ (0..^3) = {0, 1, 2}
35	33, 34	eqtri 2759	. . . . 5 ⊢ (0..^(♯‘𝑃)) = {0, 1, 2}
36	35	raleqi 3293	. . . 4 ⊢ (∀𝑘 ∈ (0..^(♯‘𝑃))(𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)) ↔ ∀𝑘 ∈ {0, 1, 2} (𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)))
37		c0ex 11138	. . . . 5 ⊢ 0 ∈ V
38		1ex 11140	. . . . 5 ⊢ 1 ∈ V
39		2ex 12258	. . . . 5 ⊢ 2 ∈ V
40		neeq1 2994	. . . . . 6 ⊢ (𝑘 = 0 → (𝑘 ≠ 1 ↔ 0 ≠ 1))
41		fveq2 6840	. . . . . . 7 ⊢ (𝑘 = 0 → (𝑃‘𝑘) = (𝑃‘0))
42	41	neeq1d 2991	. . . . . 6 ⊢ (𝑘 = 0 → ((𝑃‘𝑘) ≠ (𝑃‘1) ↔ (𝑃‘0) ≠ (𝑃‘1)))
43	40, 42	imbi12d 344	. . . . 5 ⊢ (𝑘 = 0 → ((𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)) ↔ (0 ≠ 1 → (𝑃‘0) ≠ (𝑃‘1))))
44		neeq1 2994	. . . . . 6 ⊢ (𝑘 = 1 → (𝑘 ≠ 1 ↔ 1 ≠ 1))
45		fveq2 6840	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑃‘𝑘) = (𝑃‘1))
46	45	neeq1d 2991	. . . . . 6 ⊢ (𝑘 = 1 → ((𝑃‘𝑘) ≠ (𝑃‘1) ↔ (𝑃‘1) ≠ (𝑃‘1)))
47	44, 46	imbi12d 344	. . . . 5 ⊢ (𝑘 = 1 → ((𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)) ↔ (1 ≠ 1 → (𝑃‘1) ≠ (𝑃‘1))))
48		neeq1 2994	. . . . . 6 ⊢ (𝑘 = 2 → (𝑘 ≠ 1 ↔ 2 ≠ 1))
49		fveq2 6840	. . . . . . 7 ⊢ (𝑘 = 2 → (𝑃‘𝑘) = (𝑃‘2))
50	49	neeq1d 2991	. . . . . 6 ⊢ (𝑘 = 2 → ((𝑃‘𝑘) ≠ (𝑃‘1) ↔ (𝑃‘2) ≠ (𝑃‘1)))
51	48, 50	imbi12d 344	. . . . 5 ⊢ (𝑘 = 2 → ((𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)) ↔ (2 ≠ 1 → (𝑃‘2) ≠ (𝑃‘1))))
52	37, 38, 39, 43, 47, 51	raltp 4649	. . . 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 275	. . 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 234	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝑃))(𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)))
55	3	fveq2i 6843	. . . . . . . 8 ⊢ (♯‘𝐹) = (♯‘⟨“𝐽𝐾”⟩)
56		s2len 14851	. . . . . . . 8 ⊢ (♯‘⟨“𝐽𝐾”⟩) = 2
57	55, 56	eqtri 2759	. . . . . . 7 ⊢ (♯‘𝐹) = 2
58	57	oveq2i 7378	. . . . . 6 ⊢ (1..^(♯‘𝐹)) = (1..^2)
59		fzo12sn 13703	. . . . . 6 ⊢ (1..^2) = {1}
60	58, 59	eqtri 2759	. . . . 5 ⊢ (1..^(♯‘𝐹)) = {1}
61	60	raleqi 3293	. . . 4 ⊢ (∀𝑗 ∈ (1..^(♯‘𝐹))(𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗)) ↔ ∀𝑗 ∈ {1} (𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗)))
62		neeq2 2995	. . . . . 6 ⊢ (𝑗 = 1 → (𝑘 ≠ 𝑗 ↔ 𝑘 ≠ 1))
63		fveq2 6840	. . . . . . 7 ⊢ (𝑗 = 1 → (𝑃‘𝑗) = (𝑃‘1))
64	63	neeq2d 2992	. . . . . 6 ⊢ (𝑗 = 1 → ((𝑃‘𝑘) ≠ (𝑃‘𝑗) ↔ (𝑃‘𝑘) ≠ (𝑃‘1)))
65	62, 64	imbi12d 344	. . . . 5 ⊢ (𝑗 = 1 → ((𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗)) ↔ (𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1))))
66	38, 65	ralsn 4625	. . . 4 ⊢ (∀𝑗 ∈ {1} (𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗)) ↔ (𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)))
67	61, 66	bitri 275	. . 3 ⊢ (∀𝑗 ∈ (1..^(♯‘𝐹))(𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗)) ↔ (𝑘 ≠ 1 → (𝑃‘𝑘) ≠ (𝑃‘1)))
68	67	ralbii 3083	. 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 234	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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∀wral 3051 {csn 4567 {ctp 4571 ‘cfv 6498 (class class class)co 7367 0cc0 11038 1c1 11039 2c2 12236 3c3 12237 ..^cfzo 13608 ♯chash 14292 ⟨“cs2 14803 ⟨“cs3 14804

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-z 12525 df-uz 12789 df-fz 13462 df-fzo 13609 df-hash 14293 df-word 14476 df-concat 14533 df-s1 14559 df-s2 14810 df-s3 14811

This theorem is referenced by: 2pthd 30008

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