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Theorem 2pthdlem1 30015

Description: Lemma 1 for 2pthd 30025. (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 30012	. . . 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 2994	. . . . . . . . . . 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 2737	. . . . . 6 ⊢ 1 = 1
16		eqneqall 2944	. . . . . 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 2994	. . . . . . . . . . 11 ⊢ (((𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → ((𝑃‘2) ≠ (𝑃‘1) ↔ 𝐶 ≠ 𝐵))
21		necom 2986	. . . . . . . . . . 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 6845	. . . . . . . 8 ⊢ (♯‘𝑃) = (♯‘⟨“𝐴𝐵𝐶”⟩)
31		s3len 14829	. . . . . . . 8 ⊢ (♯‘⟨“𝐴𝐵𝐶”⟩) = 3
32	30, 31	eqtri 2760	. . . . . . 7 ⊢ (♯‘𝑃) = 3
33	32	oveq2i 7379	. . . . . 6 ⊢ (0..^(♯‘𝑃)) = (0..^3)
34		fzo0to3tp 13680	. . . . . 6 ⊢ (0..^3) = {0, 1, 2}
35	33, 34	eqtri 2760	. . . . 5 ⊢ (0..^(♯‘𝑃)) = {0, 1, 2}
36	35	raleqi 3296	. . . 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 12234	. . . . 5 ⊢ 2 ∈ V
40		neeq1 2995	. . . . . 6 ⊢ (𝑘 = 0 → (𝑘 ≠ 1 ↔ 0 ≠ 1))
41		fveq2 6842	. . . . . . 7 ⊢ (𝑘 = 0 → (𝑃‘𝑘) = (𝑃‘0))
42	41	neeq1d 2992	. . . . . 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 2995	. . . . . 6 ⊢ (𝑘 = 1 → (𝑘 ≠ 1 ↔ 1 ≠ 1))
45		fveq2 6842	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑃‘𝑘) = (𝑃‘1))
46	45	neeq1d 2992	. . . . . 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 2995	. . . . . 6 ⊢ (𝑘 = 2 → (𝑘 ≠ 1 ↔ 2 ≠ 1))
49		fveq2 6842	. . . . . . 7 ⊢ (𝑘 = 2 → (𝑃‘𝑘) = (𝑃‘2))
50	49	neeq1d 2992	. . . . . 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 4664	. . . 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 6845	. . . . . . . 8 ⊢ (♯‘𝐹) = (♯‘⟨“𝐽𝐾”⟩)
56		s2len 14824	. . . . . . . 8 ⊢ (♯‘⟨“𝐽𝐾”⟩) = 2
57	55, 56	eqtri 2760	. . . . . . 7 ⊢ (♯‘𝐹) = 2
58	57	oveq2i 7379	. . . . . 6 ⊢ (1..^(♯‘𝐹)) = (1..^2)
59		fzo12sn 13676	. . . . . 6 ⊢ (1..^2) = {1}
60	58, 59	eqtri 2760	. . . . 5 ⊢ (1..^(♯‘𝐹)) = {1}
61	60	raleqi 3296	. . . 4 ⊢ (∀𝑗 ∈ (1..^(♯‘𝐹))(𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗)) ↔ ∀𝑗 ∈ {1} (𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗)))
62		neeq2 2996	. . . . . 6 ⊢ (𝑗 = 1 → (𝑘 ≠ 𝑗 ↔ 𝑘 ≠ 1))
63		fveq2 6842	. . . . . . 7 ⊢ (𝑗 = 1 → (𝑃‘𝑗) = (𝑃‘1))
64	63	neeq2d 2993	. . . . . 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 4640	. . . 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 3084	. 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 2933 ∀wral 3052 {csn 4582 {ctp 4586 ‘cfv 6500 (class class class)co 7368 0cc0 11038 1c1 11039 2c2 12212 3c3 12213 ..^cfzo 13582 ♯chash 14265 ⟨“cs2 14776 ⟨“cs3 14777

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 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-n0 12414 df-z 12501 df-uz 12764 df-fz 13436 df-fzo 13583 df-hash 14266 df-word 14449 df-concat 14506 df-s1 14532 df-s2 14783 df-s3 14784

This theorem is referenced by: 2pthd 30025

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