Theorem strleun 12589

Description: Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.)

Hypotheses

Ref	Expression
strleun.f	⊢ 𝐹 Struct ⟨𝐴, 𝐵⟩
strleun.g	⊢ 𝐺 Struct ⟨𝐶, 𝐷⟩
strleun.l	⊢ 𝐵 < 𝐶

Assertion

Ref	Expression
strleun	⊢ (𝐹 ∪ 𝐺) Struct ⟨𝐴, 𝐷⟩

Proof of Theorem strleun

Step	Hyp	Ref	Expression
1		strleun.f	. . . . . 6 ⊢ 𝐹 Struct ⟨𝐴, 𝐵⟩
2		isstructim 12501	. . . . . 6 ⊢ (𝐹 Struct ⟨𝐴, 𝐵⟩ → ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐴 ≤ 𝐵) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (𝐴...𝐵)))
3	1, 2	ax-mp 5	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐴 ≤ 𝐵) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (𝐴...𝐵))
4	3	simp1i 1008	. . . 4 ⊢ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐴 ≤ 𝐵)
5	4	simp1i 1008	. . 3 ⊢ 𝐴 ∈ ℕ
6		strleun.g	. . . . . 6 ⊢ 𝐺 Struct ⟨𝐶, 𝐷⟩
7		isstructim 12501	. . . . . 6 ⊢ (𝐺 Struct ⟨𝐶, 𝐷⟩ → ((𝐶 ∈ ℕ ∧ 𝐷 ∈ ℕ ∧ 𝐶 ≤ 𝐷) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (𝐶...𝐷)))
8	6, 7	ax-mp 5	. . . . 5 ⊢ ((𝐶 ∈ ℕ ∧ 𝐷 ∈ ℕ ∧ 𝐶 ≤ 𝐷) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (𝐶...𝐷))
9	8	simp1i 1008	. . . 4 ⊢ (𝐶 ∈ ℕ ∧ 𝐷 ∈ ℕ ∧ 𝐶 ≤ 𝐷)
10	9	simp2i 1009	. . 3 ⊢ 𝐷 ∈ ℕ
11	4	simp3i 1010	. . . . 5 ⊢ 𝐴 ≤ 𝐵
12	4	simp2i 1009	. . . . . . 7 ⊢ 𝐵 ∈ ℕ
13	12	nnrei 8948	. . . . . 6 ⊢ 𝐵 ∈ ℝ
14	9	simp1i 1008	. . . . . . 7 ⊢ 𝐶 ∈ ℕ
15	14	nnrei 8948	. . . . . 6 ⊢ 𝐶 ∈ ℝ
16		strleun.l	. . . . . 6 ⊢ 𝐵 < 𝐶
17	13, 15, 16	ltleii 8080	. . . . 5 ⊢ 𝐵 ≤ 𝐶
18	5	nnrei 8948	. . . . . 6 ⊢ 𝐴 ∈ ℝ
19	18, 13, 15	letri 8085	. . . . 5 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)
20	11, 17, 19	mp2an 426	. . . 4 ⊢ 𝐴 ≤ 𝐶
21	9	simp3i 1010	. . . 4 ⊢ 𝐶 ≤ 𝐷
22	10	nnrei 8948	. . . . 5 ⊢ 𝐷 ∈ ℝ
23	18, 15, 22	letri 8085	. . . 4 ⊢ ((𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐷) → 𝐴 ≤ 𝐷)
24	20, 21, 23	mp2an 426	. . 3 ⊢ 𝐴 ≤ 𝐷
25	5, 10, 24	3pm3.2i 1177	. 2 ⊢ (𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ∧ 𝐴 ≤ 𝐷)
26	3	simp2i 1009	. . . . . 6 ⊢ Fun (𝐹 ∖ {∅})
27	8	simp2i 1009	. . . . . 6 ⊢ Fun (𝐺 ∖ {∅})
28	26, 27	pm3.2i 272	. . . . 5 ⊢ (Fun (𝐹 ∖ {∅}) ∧ Fun (𝐺 ∖ {∅}))
29		difss 3276	. . . . . . . . 9 ⊢ (𝐹 ∖ {∅}) ⊆ 𝐹
30		dmss 4841	. . . . . . . . 9 ⊢ ((𝐹 ∖ {∅}) ⊆ 𝐹 → dom (𝐹 ∖ {∅}) ⊆ dom 𝐹)
31	29, 30	ax-mp 5	. . . . . . . 8 ⊢ dom (𝐹 ∖ {∅}) ⊆ dom 𝐹
32	3	simp3i 1010	. . . . . . . 8 ⊢ dom 𝐹 ⊆ (𝐴...𝐵)
33	31, 32	sstri 3179	. . . . . . 7 ⊢ dom (𝐹 ∖ {∅}) ⊆ (𝐴...𝐵)
34		difss 3276	. . . . . . . . 9 ⊢ (𝐺 ∖ {∅}) ⊆ 𝐺
35		dmss 4841	. . . . . . . . 9 ⊢ ((𝐺 ∖ {∅}) ⊆ 𝐺 → dom (𝐺 ∖ {∅}) ⊆ dom 𝐺)
36	34, 35	ax-mp 5	. . . . . . . 8 ⊢ dom (𝐺 ∖ {∅}) ⊆ dom 𝐺
37	8	simp3i 1010	. . . . . . . 8 ⊢ dom 𝐺 ⊆ (𝐶...𝐷)
38	36, 37	sstri 3179	. . . . . . 7 ⊢ dom (𝐺 ∖ {∅}) ⊆ (𝐶...𝐷)
39		ss2in 3378	. . . . . . 7 ⊢ ((dom (𝐹 ∖ {∅}) ⊆ (𝐴...𝐵) ∧ dom (𝐺 ∖ {∅}) ⊆ (𝐶...𝐷)) → (dom (𝐹 ∖ {∅}) ∩ dom (𝐺 ∖ {∅})) ⊆ ((𝐴...𝐵) ∩ (𝐶...𝐷)))
40	33, 38, 39	mp2an 426	. . . . . 6 ⊢ (dom (𝐹 ∖ {∅}) ∩ dom (𝐺 ∖ {∅})) ⊆ ((𝐴...𝐵) ∩ (𝐶...𝐷))
41		fzdisj 10072	. . . . . . 7 ⊢ (𝐵 < 𝐶 → ((𝐴...𝐵) ∩ (𝐶...𝐷)) = ∅)
42	16, 41	ax-mp 5	. . . . . 6 ⊢ ((𝐴...𝐵) ∩ (𝐶...𝐷)) = ∅
43		sseq0 3479	. . . . . 6 ⊢ (((dom (𝐹 ∖ {∅}) ∩ dom (𝐺 ∖ {∅})) ⊆ ((𝐴...𝐵) ∩ (𝐶...𝐷)) ∧ ((𝐴...𝐵) ∩ (𝐶...𝐷)) = ∅) → (dom (𝐹 ∖ {∅}) ∩ dom (𝐺 ∖ {∅})) = ∅)
44	40, 42, 43	mp2an 426	. . . . 5 ⊢ (dom (𝐹 ∖ {∅}) ∩ dom (𝐺 ∖ {∅})) = ∅
45		funun 5276	. . . . 5 ⊢ (((Fun (𝐹 ∖ {∅}) ∧ Fun (𝐺 ∖ {∅})) ∧ (dom (𝐹 ∖ {∅}) ∩ dom (𝐺 ∖ {∅})) = ∅) → Fun ((𝐹 ∖ {∅}) ∪ (𝐺 ∖ {∅})))
46	28, 44, 45	mp2an 426	. . . 4 ⊢ Fun ((𝐹 ∖ {∅}) ∪ (𝐺 ∖ {∅}))
47		difundir 3403	. . . . 5 ⊢ ((𝐹 ∪ 𝐺) ∖ {∅}) = ((𝐹 ∖ {∅}) ∪ (𝐺 ∖ {∅}))
48	47	funeqi 5253	. . . 4 ⊢ (Fun ((𝐹 ∪ 𝐺) ∖ {∅}) ↔ Fun ((𝐹 ∖ {∅}) ∪ (𝐺 ∖ {∅})))
49	46, 48	mpbir 146	. . 3 ⊢ Fun ((𝐹 ∪ 𝐺) ∖ {∅})
50		structex 12499	. . . . 5 ⊢ (𝐹 Struct ⟨𝐴, 𝐵⟩ → 𝐹 ∈ V)
51	1, 50	ax-mp 5	. . . 4 ⊢ 𝐹 ∈ V
52		structex 12499	. . . . 5 ⊢ (𝐺 Struct ⟨𝐶, 𝐷⟩ → 𝐺 ∈ V)
53	6, 52	ax-mp 5	. . . 4 ⊢ 𝐺 ∈ V
54	51, 53	unex 4456	. . 3 ⊢ (𝐹 ∪ 𝐺) ∈ V
55		dmun 4849	. . . 4 ⊢ dom (𝐹 ∪ 𝐺) = (dom 𝐹 ∪ dom 𝐺)
56	12	nnzi 9294	. . . . . . . 8 ⊢ 𝐵 ∈ ℤ
57	10	nnzi 9294	. . . . . . . 8 ⊢ 𝐷 ∈ ℤ
58	13, 15, 22	letri 8085	. . . . . . . . 9 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐷) → 𝐵 ≤ 𝐷)
59	17, 21, 58	mp2an 426	. . . . . . . 8 ⊢ 𝐵 ≤ 𝐷
60		eluz2 9554	. . . . . . . 8 ⊢ (𝐷 ∈ (ℤ≥‘𝐵) ↔ (𝐵 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐵 ≤ 𝐷))
61	56, 57, 59, 60	mpbir3an 1181	. . . . . . 7 ⊢ 𝐷 ∈ (ℤ≥‘𝐵)
62		fzss2 10084	. . . . . . 7 ⊢ (𝐷 ∈ (ℤ≥‘𝐵) → (𝐴...𝐵) ⊆ (𝐴...𝐷))
63	61, 62	ax-mp 5	. . . . . 6 ⊢ (𝐴...𝐵) ⊆ (𝐴...𝐷)
64	32, 63	sstri 3179	. . . . 5 ⊢ dom 𝐹 ⊆ (𝐴...𝐷)
65	5	nnzi 9294	. . . . . . . 8 ⊢ 𝐴 ∈ ℤ
66	14	nnzi 9294	. . . . . . . 8 ⊢ 𝐶 ∈ ℤ
67		eluz2 9554	. . . . . . . 8 ⊢ (𝐶 ∈ (ℤ≥‘𝐴) ↔ (𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐴 ≤ 𝐶))
68	65, 66, 20, 67	mpbir3an 1181	. . . . . . 7 ⊢ 𝐶 ∈ (ℤ≥‘𝐴)
69		fzss1 10083	. . . . . . 7 ⊢ (𝐶 ∈ (ℤ≥‘𝐴) → (𝐶...𝐷) ⊆ (𝐴...𝐷))
70	68, 69	ax-mp 5	. . . . . 6 ⊢ (𝐶...𝐷) ⊆ (𝐴...𝐷)
71	37, 70	sstri 3179	. . . . 5 ⊢ dom 𝐺 ⊆ (𝐴...𝐷)
72	64, 71	unssi 3325	. . . 4 ⊢ (dom 𝐹 ∪ dom 𝐺) ⊆ (𝐴...𝐷)
73	55, 72	eqsstri 3202	. . 3 ⊢ dom (𝐹 ∪ 𝐺) ⊆ (𝐴...𝐷)
74	49, 54, 73	3pm3.2i 1177	. 2 ⊢ (Fun ((𝐹 ∪ 𝐺) ∖ {∅}) ∧ (𝐹 ∪ 𝐺) ∈ V ∧ dom (𝐹 ∪ 𝐺) ⊆ (𝐴...𝐷))
75		isstructr 12502	. 2 ⊢ (((𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ∧ 𝐴 ≤ 𝐷) ∧ (Fun ((𝐹 ∪ 𝐺) ∖ {∅}) ∧ (𝐹 ∪ 𝐺) ∈ V ∧ dom (𝐹 ∪ 𝐺) ⊆ (𝐴...𝐷))) → (𝐹 ∪ 𝐺) Struct ⟨𝐴, 𝐷⟩)
76	25, 74, 75	mp2an 426	1 ⊢ (𝐹 ∪ 𝐺) Struct ⟨𝐴, 𝐷⟩

Syntax hints:   ∧ wa 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2160  Vcvv 2752   ∖ cdif 3141   ∪ cun 3142   ∩ cin 3143   ⊆ wss 3144  ∅c0 3437  {csn 3607  ⟨cop 3610   class class class wbr 4018  dom cdm 4641  Fun wfun 5226  ‘cfv 5232  (class class class)co 5892   < clt 8012   ≤ cle 8013  ℕcn 8939  ℤcz 9273  ℤ_≥cuz 9548  ...cfz 10028   Struct cstr 12483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7922  ax-resscn 7923  ax-1cn 7924  ax-1re 7925  ax-icn 7926  ax-addcl 7927  ax-addrcl 7928  ax-mulcl 7929  ax-addcom 7931  ax-addass 7933  ax-distr 7935  ax-i2m1 7936  ax-0lt1 7937  ax-0id 7939  ax-rnegex 7940  ax-cnre 7942  ax-pre-ltirr 7943  ax-pre-ltwlin 7944  ax-pre-lttrn 7945  ax-pre-ltadd 7947

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-fv 5240  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-pnf 8014  df-mnf 8015  df-xr 8016  df-ltxr 8017  df-le 8018  df-sub 8150  df-neg 8151  df-inn 8940  df-z 9274  df-uz 9549  df-fz 10029  df-struct 12489

This theorem is referenced by: (None)