Theorem strleun 12782

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 12692	. . . . . 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 12692	. . . . . 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 8999	. . . . . 6 ⊢ 𝐵 ∈ ℝ
14	9	simp1i 1008	. . . . . . 7 ⊢ 𝐶 ∈ ℕ
15	14	nnrei 8999	. . . . . 6 ⊢ 𝐶 ∈ ℝ
16		strleun.l	. . . . . 6 ⊢ 𝐵 < 𝐶
17	13, 15, 16	ltleii 8129	. . . . 5 ⊢ 𝐵 ≤ 𝐶
18	5	nnrei 8999	. . . . . 6 ⊢ 𝐴 ∈ ℝ
19	18, 13, 15	letri 8134	. . . . 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 8999	. . . . 5 ⊢ 𝐷 ∈ ℝ
23	18, 15, 22	letri 8134	. . . 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 3289	. . . . . . . . 9 ⊢ (𝐹 ∖ {∅}) ⊆ 𝐹
30		dmss 4865	. . . . . . . . 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 3192	. . . . . . 7 ⊢ dom (𝐹 ∖ {∅}) ⊆ (𝐴...𝐵)
34		difss 3289	. . . . . . . . 9 ⊢ (𝐺 ∖ {∅}) ⊆ 𝐺
35		dmss 4865	. . . . . . . . 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 3192	. . . . . . 7 ⊢ dom (𝐺 ∖ {∅}) ⊆ (𝐶...𝐷)
39		ss2in 3391	. . . . . . 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 10127	. . . . . . 7 ⊢ (𝐵 < 𝐶 → ((𝐴...𝐵) ∩ (𝐶...𝐷)) = ∅)
42	16, 41	ax-mp 5	. . . . . 6 ⊢ ((𝐴...𝐵) ∩ (𝐶...𝐷)) = ∅
43		sseq0 3492	. . . . . 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 5302	. . . . 5 ⊢ (((Fun (𝐹 ∖ {∅}) ∧ Fun (𝐺 ∖ {∅})) ∧ (dom (𝐹 ∖ {∅}) ∩ dom (𝐺 ∖ {∅})) = ∅) → Fun ((𝐹 ∖ {∅}) ∪ (𝐺 ∖ {∅})))
46	28, 44, 45	mp2an 426	. . . 4 ⊢ Fun ((𝐹 ∖ {∅}) ∪ (𝐺 ∖ {∅}))
47		difundir 3416	. . . . 5 ⊢ ((𝐹 ∪ 𝐺) ∖ {∅}) = ((𝐹 ∖ {∅}) ∪ (𝐺 ∖ {∅}))
48	47	funeqi 5279	. . . 4 ⊢ (Fun ((𝐹 ∪ 𝐺) ∖ {∅}) ↔ Fun ((𝐹 ∖ {∅}) ∪ (𝐺 ∖ {∅})))
49	46, 48	mpbir 146	. . 3 ⊢ Fun ((𝐹 ∪ 𝐺) ∖ {∅})
50		structex 12690	. . . . 5 ⊢ (𝐹 Struct ⟨𝐴, 𝐵⟩ → 𝐹 ∈ V)
51	1, 50	ax-mp 5	. . . 4 ⊢ 𝐹 ∈ V
52		structex 12690	. . . . 5 ⊢ (𝐺 Struct ⟨𝐶, 𝐷⟩ → 𝐺 ∈ V)
53	6, 52	ax-mp 5	. . . 4 ⊢ 𝐺 ∈ V
54	51, 53	unex 4476	. . 3 ⊢ (𝐹 ∪ 𝐺) ∈ V
55		dmun 4873	. . . 4 ⊢ dom (𝐹 ∪ 𝐺) = (dom 𝐹 ∪ dom 𝐺)
56	12	nnzi 9347	. . . . . . . 8 ⊢ 𝐵 ∈ ℤ
57	10	nnzi 9347	. . . . . . . 8 ⊢ 𝐷 ∈ ℤ
58	13, 15, 22	letri 8134	. . . . . . . . 9 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐷) → 𝐵 ≤ 𝐷)
59	17, 21, 58	mp2an 426	. . . . . . . 8 ⊢ 𝐵 ≤ 𝐷
60		eluz2 9607	. . . . . . . 8 ⊢ (𝐷 ∈ (ℤ≥‘𝐵) ↔ (𝐵 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐵 ≤ 𝐷))
61	56, 57, 59, 60	mpbir3an 1181	. . . . . . 7 ⊢ 𝐷 ∈ (ℤ≥‘𝐵)
62		fzss2 10139	. . . . . . 7 ⊢ (𝐷 ∈ (ℤ≥‘𝐵) → (𝐴...𝐵) ⊆ (𝐴...𝐷))
63	61, 62	ax-mp 5	. . . . . 6 ⊢ (𝐴...𝐵) ⊆ (𝐴...𝐷)
64	32, 63	sstri 3192	. . . . 5 ⊢ dom 𝐹 ⊆ (𝐴...𝐷)
65	5	nnzi 9347	. . . . . . . 8 ⊢ 𝐴 ∈ ℤ
66	14	nnzi 9347	. . . . . . . 8 ⊢ 𝐶 ∈ ℤ
67		eluz2 9607	. . . . . . . 8 ⊢ (𝐶 ∈ (ℤ≥‘𝐴) ↔ (𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐴 ≤ 𝐶))
68	65, 66, 20, 67	mpbir3an 1181	. . . . . . 7 ⊢ 𝐶 ∈ (ℤ≥‘𝐴)
69		fzss1 10138	. . . . . . 7 ⊢ (𝐶 ∈ (ℤ≥‘𝐴) → (𝐶...𝐷) ⊆ (𝐴...𝐷))
70	68, 69	ax-mp 5	. . . . . 6 ⊢ (𝐶...𝐷) ⊆ (𝐴...𝐷)
71	37, 70	sstri 3192	. . . . 5 ⊢ dom 𝐺 ⊆ (𝐴...𝐷)
72	64, 71	unssi 3338	. . . 4 ⊢ (dom 𝐹 ∪ dom 𝐺) ⊆ (𝐴...𝐷)
73	55, 72	eqsstri 3215	. . 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 12693	. 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 2167  Vcvv 2763   ∖ cdif 3154   ∪ cun 3155   ∩ cin 3156   ⊆ wss 3157  ∅c0 3450  {csn 3622  ⟨cop 3625   class class class wbr 4033  dom cdm 4663  Fun wfun 5252  ‘cfv 5258  (class class class)co 5922   < clt 8061   ≤ cle 8062  ℕcn 8990  ℤcz 9326  ℤ_≥cuz 9601  ...cfz 10083   Struct cstr 12674

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-z 9327  df-uz 9602  df-fz 10084  df-struct 12680

This theorem is referenced by:  cnfldstr  14114