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Theorem constr3trllem3 26180

Description: Lemma for constr3trl 26187. (Contributed by Alexander van der Vekens, 11-Nov-2017.)

**Hypotheses**
Ref	Expression
constr3cycl.f	⊢ 𝐹 = {⟨0, (^◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (^◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (^◡𝐸‘{𝐶, 𝐴})⟩}
constr3cycl.p	⊢ 𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})

**Assertion**
Ref	Expression
constr3trllem3	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝑃:(0...(#‘𝐹))⟶𝑉)

**Proof of Theorem constr3trllem3**
Step	Hyp	Ref	Expression
1		0z 11265	. . . . . 6 ⊢ 0 ∈ ℤ
2		1z 11284	. . . . . 6 ⊢ 1 ∈ ℤ
3	1, 2	pm3.2i 470	. . . . 5 ⊢ (0 ∈ ℤ ∧ 1 ∈ ℤ)
4	3	a1i 11	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (0 ∈ ℤ ∧ 1 ∈ ℤ))
5		3simpa 1051	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))
6		0ne1 10965	. . . . 5 ⊢ 0 ≠ 1
7	6	a1i 11	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 0 ≠ 1)
8		fprg 6327	. . . . 5 ⊢ (((0 ∈ ℤ ∧ 1 ∈ ℤ) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 0 ≠ 1) → {⟨0, 𝐴⟩, ⟨1, 𝐵⟩}:{0, 1}⟶{𝐴, 𝐵})
9		0p1e1 11009	. . . . . . . . 9 ⊢ (0 + 1) = 1
10	9	eqcomi 2619	. . . . . . . 8 ⊢ 1 = (0 + 1)
11	10	oveq2i 6560	. . . . . . 7 ⊢ (0...1) = (0...(0 + 1))
12		fzpr 12266	. . . . . . . 8 ⊢ (0 ∈ ℤ → (0...(0 + 1)) = {0, (0 + 1)})
13	1, 12	ax-mp 5	. . . . . . 7 ⊢ (0...(0 + 1)) = {0, (0 + 1)}
14	9	preq2i 4216	. . . . . . 7 ⊢ {0, (0 + 1)} = {0, 1}
15	11, 13, 14	3eqtri 2636	. . . . . 6 ⊢ (0...1) = {0, 1}
16	15	feq2i 5950	. . . . 5 ⊢ ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩}:(0...1)⟶{𝐴, 𝐵} ↔ {⟨0, 𝐴⟩, ⟨1, 𝐵⟩}:{0, 1}⟶{𝐴, 𝐵})
17	8, 16	sylibr 223	. . . 4 ⊢ (((0 ∈ ℤ ∧ 1 ∈ ℤ) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 0 ≠ 1) → {⟨0, 𝐴⟩, ⟨1, 𝐵⟩}:(0...1)⟶{𝐴, 𝐵})
18	4, 5, 7, 17	syl3anc 1318	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → {⟨0, 𝐴⟩, ⟨1, 𝐵⟩}:(0...1)⟶{𝐴, 𝐵})
19		prssi 4293	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → {𝐴, 𝐵} ⊆ 𝑉)
20	19	3adant3 1074	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → {𝐴, 𝐵} ⊆ 𝑉)
21	18, 20	fssd 5970	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → {⟨0, 𝐴⟩, ⟨1, 𝐵⟩}:(0...1)⟶𝑉)
22		2nn0 11186	. . . . . 6 ⊢ 2 ∈ ℕ₀
23		3nn0 11187	. . . . . 6 ⊢ 3 ∈ ℕ₀
24	22, 23	pm3.2i 470	. . . . 5 ⊢ (2 ∈ ℕ₀ ∧ 3 ∈ ℕ₀)
25	24	a1i 11	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (2 ∈ ℕ₀ ∧ 3 ∈ ℕ₀))
26		pm3.22 464	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉))
27	26	3adant2 1073	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉))
28		2re 10967	. . . . . 6 ⊢ 2 ∈ ℝ
29		2lt3 11072	. . . . . 6 ⊢ 2 < 3
30	28, 29	ltneii 10029	. . . . 5 ⊢ 2 ≠ 3
31	30	a1i 11	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 2 ≠ 3)
32		fprg 6327	. . . . 5 ⊢ (((2 ∈ ℕ₀ ∧ 3 ∈ ℕ₀) ∧ (𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ 2 ≠ 3) → {⟨2, 𝐶⟩, ⟨3, 𝐴⟩}:{2, 3}⟶{𝐶, 𝐴})
33		constr3cycl.f	. . . . . . . . . 10 ⊢ 𝐹 = {⟨0, (^◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (^◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (^◡𝐸‘{𝐶, 𝐴})⟩}
34		constr3cycl.p	. . . . . . . . . 10 ⊢ 𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})
35	33, 34	constr3lem2 26174	. . . . . . . . 9 ⊢ (#‘𝐹) = 3
36		df-3 10957	. . . . . . . . 9 ⊢ 3 = (2 + 1)
37	35, 36	eqtri 2632	. . . . . . . 8 ⊢ (#‘𝐹) = (2 + 1)
38	37	oveq2i 6560	. . . . . . 7 ⊢ (2...(#‘𝐹)) = (2...(2 + 1))
39		2z 11286	. . . . . . . 8 ⊢ 2 ∈ ℤ
40		fzpr 12266	. . . . . . . 8 ⊢ (2 ∈ ℤ → (2...(2 + 1)) = {2, (2 + 1)})
41	39, 40	ax-mp 5	. . . . . . 7 ⊢ (2...(2 + 1)) = {2, (2 + 1)}
42	36	eqcomi 2619	. . . . . . . 8 ⊢ (2 + 1) = 3
43	42	preq2i 4216	. . . . . . 7 ⊢ {2, (2 + 1)} = {2, 3}
44	38, 41, 43	3eqtri 2636	. . . . . 6 ⊢ (2...(#‘𝐹)) = {2, 3}
45	44	feq2i 5950	. . . . 5 ⊢ ({⟨2, 𝐶⟩, ⟨3, 𝐴⟩}:(2...(#‘𝐹))⟶{𝐶, 𝐴} ↔ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩}:{2, 3}⟶{𝐶, 𝐴})
46	32, 45	sylibr 223	. . . 4 ⊢ (((2 ∈ ℕ₀ ∧ 3 ∈ ℕ₀) ∧ (𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) ∧ 2 ≠ 3) → {⟨2, 𝐶⟩, ⟨3, 𝐴⟩}:(2...(#‘𝐹))⟶{𝐶, 𝐴})
47	25, 27, 31, 46	syl3anc 1318	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → {⟨2, 𝐶⟩, ⟨3, 𝐴⟩}:(2...(#‘𝐹))⟶{𝐶, 𝐴})
48		prssi 4293	. . . . 5 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → {𝐶, 𝐴} ⊆ 𝑉)
49	48	ancoms 468	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → {𝐶, 𝐴} ⊆ 𝑉)
50	49	3adant2 1073	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → {𝐶, 𝐴} ⊆ 𝑉)
51	47, 50	fssd 5970	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → {⟨2, 𝐶⟩, ⟨3, 𝐴⟩}:(2...(#‘𝐹))⟶𝑉)
52		1lt2 11071	. . 3 ⊢ 1 < 2
53		fzdisj 12239	. . 3 ⊢ (1 < 2 → ((0...1) ∩ (2...(#‘𝐹))) = ∅)
54	52, 53	mp1i 13	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((0...1) ∩ (2...(#‘𝐹))) = ∅)
55		fun 5979	. . 3 ⊢ ((({⟨0, 𝐴⟩, ⟨1, 𝐵⟩}:(0...1)⟶𝑉 ∧ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩}:(2...(#‘𝐹))⟶𝑉) ∧ ((0...1) ∩ (2...(#‘𝐹))) = ∅) → ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩}):((0...1) ∪ (2...(#‘𝐹)))⟶(𝑉 ∪ 𝑉))
56	34	a1i 11	. . . 4 ⊢ ((({⟨0, 𝐴⟩, ⟨1, 𝐵⟩}:(0...1)⟶𝑉 ∧ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩}:(2...(#‘𝐹))⟶𝑉) ∧ ((0...1) ∩ (2...(#‘𝐹))) = ∅) → 𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩}))
57		3nn 11063	. . . . . . 7 ⊢ 3 ∈ ℕ
58	35, 57	eqeltri 2684	. . . . . 6 ⊢ (#‘𝐹) ∈ ℕ
59		elnnuz 11600	. . . . . 6 ⊢ ((#‘𝐹) ∈ ℕ ↔ (#‘𝐹) ∈ (ℤ_≥‘1))
60	58, 59	mpbi 219	. . . . 5 ⊢ (#‘𝐹) ∈ (ℤ_≥‘1)
61		0le1 10430	. . . . . . . . 9 ⊢ 0 ≤ 1
62	61	a1i 11	. . . . . . . 8 ⊢ ((#‘𝐹) ∈ (ℤ_≥‘1) → 0 ≤ 1)
63		eluzle 11576	. . . . . . . 8 ⊢ ((#‘𝐹) ∈ (ℤ_≥‘1) → 1 ≤ (#‘𝐹))
64		eluzelz 11573	. . . . . . . . 9 ⊢ ((#‘𝐹) ∈ (ℤ_≥‘1) → (#‘𝐹) ∈ ℤ)
65		elfz 12203	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ 0 ∈ ℤ ∧ (#‘𝐹) ∈ ℤ) → (1 ∈ (0...(#‘𝐹)) ↔ (0 ≤ 1 ∧ 1 ≤ (#‘𝐹))))
66	2, 1, 65	mp3an12 1406	. . . . . . . . 9 ⊢ ((#‘𝐹) ∈ ℤ → (1 ∈ (0...(#‘𝐹)) ↔ (0 ≤ 1 ∧ 1 ≤ (#‘𝐹))))
67	64, 66	syl 17	. . . . . . . 8 ⊢ ((#‘𝐹) ∈ (ℤ_≥‘1) → (1 ∈ (0...(#‘𝐹)) ↔ (0 ≤ 1 ∧ 1 ≤ (#‘𝐹))))
68	62, 63, 67	mpbir2and 959	. . . . . . 7 ⊢ ((#‘𝐹) ∈ (ℤ_≥‘1) → 1 ∈ (0...(#‘𝐹)))
69		fzsplit 12238	. . . . . . 7 ⊢ (1 ∈ (0...(#‘𝐹)) → (0...(#‘𝐹)) = ((0...1) ∪ ((1 + 1)...(#‘𝐹))))
70	68, 69	syl 17	. . . . . 6 ⊢ ((#‘𝐹) ∈ (ℤ_≥‘1) → (0...(#‘𝐹)) = ((0...1) ∪ ((1 + 1)...(#‘𝐹))))
71		df-2 10956	. . . . . . . 8 ⊢ 2 = (1 + 1)
72	71	oveq1i 6559	. . . . . . 7 ⊢ (2...(#‘𝐹)) = ((1 + 1)...(#‘𝐹))
73	72	uneq2i 3726	. . . . . 6 ⊢ ((0...1) ∪ (2...(#‘𝐹))) = ((0...1) ∪ ((1 + 1)...(#‘𝐹)))
74	70, 73	syl6eqr 2662	. . . . 5 ⊢ ((#‘𝐹) ∈ (ℤ_≥‘1) → (0...(#‘𝐹)) = ((0...1) ∪ (2...(#‘𝐹))))
75	60, 74	mp1i 13	. . . 4 ⊢ ((({⟨0, 𝐴⟩, ⟨1, 𝐵⟩}:(0...1)⟶𝑉 ∧ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩}:(2...(#‘𝐹))⟶𝑉) ∧ ((0...1) ∩ (2...(#‘𝐹))) = ∅) → (0...(#‘𝐹)) = ((0...1) ∪ (2...(#‘𝐹))))
76		unidm 3718	. . . . . 6 ⊢ (𝑉 ∪ 𝑉) = 𝑉
77	76	eqcomi 2619	. . . . 5 ⊢ 𝑉 = (𝑉 ∪ 𝑉)
78	77	a1i 11	. . . 4 ⊢ ((({⟨0, 𝐴⟩, ⟨1, 𝐵⟩}:(0...1)⟶𝑉 ∧ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩}:(2...(#‘𝐹))⟶𝑉) ∧ ((0...1) ∩ (2...(#‘𝐹))) = ∅) → 𝑉 = (𝑉 ∪ 𝑉))
79	56, 75, 78	feq123d 5947	. . 3 ⊢ ((({⟨0, 𝐴⟩, ⟨1, 𝐵⟩}:(0...1)⟶𝑉 ∧ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩}:(2...(#‘𝐹))⟶𝑉) ∧ ((0...1) ∩ (2...(#‘𝐹))) = ∅) → (𝑃:(0...(#‘𝐹))⟶𝑉 ↔ ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩}):((0...1) ∪ (2...(#‘𝐹)))⟶(𝑉 ∪ 𝑉)))
80	55, 79	mpbird 246	. 2 ⊢ ((({⟨0, 𝐴⟩, ⟨1, 𝐵⟩}:(0...1)⟶𝑉 ∧ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩}:(2...(#‘𝐹))⟶𝑉) ∧ ((0...1) ∩ (2...(#‘𝐹))) = ∅) → 𝑃:(0...(#‘𝐹))⟶𝑉)
81	21, 51, 54, 80	syl21anc 1317	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝑃:(0...(#‘𝐹))⟶𝑉)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {cpr 4127 {ctp 4129 ⟨cop 4131 class class class wbr 4583 ^◡ccnv 5037 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 ℕcn 10897 2c2 10947 3c3 10948 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ...cfz 12197 #chash 12979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980

This theorem is referenced by: constr3trllem4 26181 constr3trl 26187

W3C validator