Theorem strleun 13511

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 13418	. . . . . 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 1037	. . . 4 ⊢ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐴 ≤ 𝐵)
5	4	simp1i 1037	. . 3 ⊢ 𝐴 ∈ ℕ
6		strleun.g	. . . . . 6 ⊢ 𝐺 Struct ⟨𝐶, 𝐷⟩
7		isstructim 13418	. . . . . 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 1037	. . . 4 ⊢ (𝐶 ∈ ℕ ∧ 𝐷 ∈ ℕ ∧ 𝐶 ≤ 𝐷)
10	9	simp2i 1038	. . 3 ⊢ 𝐷 ∈ ℕ
11	4	simp3i 1039	. . . . 5 ⊢ 𝐴 ≤ 𝐵
12	4	simp2i 1038	. . . . . . 7 ⊢ 𝐵 ∈ ℕ
13	12	nnrei 9316	. . . . . 6 ⊢ 𝐵 ∈ ℝ
14	9	simp1i 1037	. . . . . . 7 ⊢ 𝐶 ∈ ℕ
15	14	nnrei 9316	. . . . . 6 ⊢ 𝐶 ∈ ℝ
16		strleun.l	. . . . . 6 ⊢ 𝐵 < 𝐶
17	13, 15, 16	ltleii 8430	. . . . 5 ⊢ 𝐵 ≤ 𝐶
18	5	nnrei 9316	. . . . . 6 ⊢ 𝐴 ∈ ℝ
19	18, 13, 15	letri 8435	. . . . 5 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)
20	11, 17, 19	mp2an 430	. . . 4 ⊢ 𝐴 ≤ 𝐶
21	9	simp3i 1039	. . . 4 ⊢ 𝐶 ≤ 𝐷
22	10	nnrei 9316	. . . . 5 ⊢ 𝐷 ∈ ℝ
23	18, 15, 22	letri 8435	. . . 4 ⊢ ((𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐷) → 𝐴 ≤ 𝐷)
24	20, 21, 23	mp2an 430	. . 3 ⊢ 𝐴 ≤ 𝐷
25	5, 10, 24	3pm3.2i 1206	. 2 ⊢ (𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ∧ 𝐴 ≤ 𝐷)
26	3	simp2i 1038	. . . . . 6 ⊢ Fun (𝐹 ∖ {∅})
27	8	simp2i 1038	. . . . . 6 ⊢ Fun (𝐺 ∖ {∅})
28	26, 27	pm3.2i 272	. . . . 5 ⊢ (Fun (𝐹 ∖ {∅}) ∧ Fun (𝐺 ∖ {∅}))
29		difss 3355	. . . . . . . . 9 ⊢ (𝐹 ∖ {∅}) ⊆ 𝐹
30		dmss 4980	. . . . . . . . 9 ⊢ ((𝐹 ∖ {∅}) ⊆ 𝐹 → dom (𝐹 ∖ {∅}) ⊆ dom 𝐹)
31	29, 30	ax-mp 5	. . . . . . . 8 ⊢ dom (𝐹 ∖ {∅}) ⊆ dom 𝐹
32	3	simp3i 1039	. . . . . . . 8 ⊢ dom 𝐹 ⊆ (𝐴...𝐵)
33	31, 32	sstri 3257	. . . . . . 7 ⊢ dom (𝐹 ∖ {∅}) ⊆ (𝐴...𝐵)
34		difss 3355	. . . . . . . . 9 ⊢ (𝐺 ∖ {∅}) ⊆ 𝐺
35		dmss 4980	. . . . . . . . 9 ⊢ ((𝐺 ∖ {∅}) ⊆ 𝐺 → dom (𝐺 ∖ {∅}) ⊆ dom 𝐺)
36	34, 35	ax-mp 5	. . . . . . . 8 ⊢ dom (𝐺 ∖ {∅}) ⊆ dom 𝐺
37	8	simp3i 1039	. . . . . . . 8 ⊢ dom 𝐺 ⊆ (𝐶...𝐷)
38	36, 37	sstri 3257	. . . . . . 7 ⊢ dom (𝐺 ∖ {∅}) ⊆ (𝐶...𝐷)
39		ss2in 3459	. . . . . . 7 ⊢ ((dom (𝐹 ∖ {∅}) ⊆ (𝐴...𝐵) ∧ dom (𝐺 ∖ {∅}) ⊆ (𝐶...𝐷)) → (dom (𝐹 ∖ {∅}) ∩ dom (𝐺 ∖ {∅})) ⊆ ((𝐴...𝐵) ∩ (𝐶...𝐷)))
40	33, 38, 39	mp2an 430	. . . . . 6 ⊢ (dom (𝐹 ∖ {∅}) ∩ dom (𝐺 ∖ {∅})) ⊆ ((𝐴...𝐵) ∩ (𝐶...𝐷))
41		fzdisj 10468	. . . . . . 7 ⊢ (𝐵 < 𝐶 → ((𝐴...𝐵) ∩ (𝐶...𝐷)) = ∅)
42	16, 41	ax-mp 5	. . . . . 6 ⊢ ((𝐴...𝐵) ∩ (𝐶...𝐷)) = ∅
43		sseq0 3565	. . . . . 6 ⊢ (((dom (𝐹 ∖ {∅}) ∩ dom (𝐺 ∖ {∅})) ⊆ ((𝐴...𝐵) ∩ (𝐶...𝐷)) ∧ ((𝐴...𝐵) ∩ (𝐶...𝐷)) = ∅) → (dom (𝐹 ∖ {∅}) ∩ dom (𝐺 ∖ {∅})) = ∅)
44	40, 42, 43	mp2an 430	. . . . 5 ⊢ (dom (𝐹 ∖ {∅}) ∩ dom (𝐺 ∖ {∅})) = ∅
45		funun 5422	. . . . 5 ⊢ (((Fun (𝐹 ∖ {∅}) ∧ Fun (𝐺 ∖ {∅})) ∧ (dom (𝐹 ∖ {∅}) ∩ dom (𝐺 ∖ {∅})) = ∅) → Fun ((𝐹 ∖ {∅}) ∪ (𝐺 ∖ {∅})))
46	28, 44, 45	mp2an 430	. . . 4 ⊢ Fun ((𝐹 ∖ {∅}) ∪ (𝐺 ∖ {∅}))
47		difundir 3484	. . . . 5 ⊢ ((𝐹 ∪ 𝐺) ∖ {∅}) = ((𝐹 ∖ {∅}) ∪ (𝐺 ∖ {∅}))
48	47	funeqi 5398	. . . 4 ⊢ (Fun ((𝐹 ∪ 𝐺) ∖ {∅}) ↔ Fun ((𝐹 ∖ {∅}) ∪ (𝐺 ∖ {∅})))
49	46, 48	mpbir 146	. . 3 ⊢ Fun ((𝐹 ∪ 𝐺) ∖ {∅})
50		structex 13416	. . . . 5 ⊢ (𝐹 Struct ⟨𝐴, 𝐵⟩ → 𝐹 ∈ V)
51	1, 50	ax-mp 5	. . . 4 ⊢ 𝐹 ∈ V
52		structex 13416	. . . . 5 ⊢ (𝐺 Struct ⟨𝐶, 𝐷⟩ → 𝐺 ∈ V)
53	6, 52	ax-mp 5	. . . 4 ⊢ 𝐺 ∈ V
54	51, 53	unex 4587	. . 3 ⊢ (𝐹 ∪ 𝐺) ∈ V
55		dmun 4988	. . . 4 ⊢ dom (𝐹 ∪ 𝐺) = (dom 𝐹 ∪ dom 𝐺)
56	12	nnzi 9670	. . . . . . . 8 ⊢ 𝐵 ∈ ℤ
57	10	nnzi 9670	. . . . . . . 8 ⊢ 𝐷 ∈ ℤ
58	13, 15, 22	letri 8435	. . . . . . . . 9 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐷) → 𝐵 ≤ 𝐷)
59	17, 21, 58	mp2an 430	. . . . . . . 8 ⊢ 𝐵 ≤ 𝐷
60		eluz2 9937	. . . . . . . 8 ⊢ (𝐷 ∈ (ℤ≥‘𝐵) ↔ (𝐵 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐵 ≤ 𝐷))
61	56, 57, 59, 60	mpbir3an 1210	. . . . . . 7 ⊢ 𝐷 ∈ (ℤ≥‘𝐵)
62		fzss2 10481	. . . . . . 7 ⊢ (𝐷 ∈ (ℤ≥‘𝐵) → (𝐴...𝐵) ⊆ (𝐴...𝐷))
63	61, 62	ax-mp 5	. . . . . 6 ⊢ (𝐴...𝐵) ⊆ (𝐴...𝐷)
64	32, 63	sstri 3257	. . . . 5 ⊢ dom 𝐹 ⊆ (𝐴...𝐷)
65	5	nnzi 9670	. . . . . . . 8 ⊢ 𝐴 ∈ ℤ
66	14	nnzi 9670	. . . . . . . 8 ⊢ 𝐶 ∈ ℤ
67		eluz2 9937	. . . . . . . 8 ⊢ (𝐶 ∈ (ℤ≥‘𝐴) ↔ (𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐴 ≤ 𝐶))
68	65, 66, 20, 67	mpbir3an 1210	. . . . . . 7 ⊢ 𝐶 ∈ (ℤ≥‘𝐴)
69		fzss1 10480	. . . . . . 7 ⊢ (𝐶 ∈ (ℤ≥‘𝐴) → (𝐶...𝐷) ⊆ (𝐴...𝐷))
70	68, 69	ax-mp 5	. . . . . 6 ⊢ (𝐶...𝐷) ⊆ (𝐴...𝐷)
71	37, 70	sstri 3257	. . . . 5 ⊢ dom 𝐺 ⊆ (𝐴...𝐷)
72	64, 71	unssi 3404	. . . 4 ⊢ (dom 𝐹 ∪ dom 𝐺) ⊆ (𝐴...𝐷)
73	55, 72	eqsstri 3280	. . 3 ⊢ dom (𝐹 ∪ 𝐺) ⊆ (𝐴...𝐷)
74	49, 54, 73	3pm3.2i 1206	. 2 ⊢ (Fun ((𝐹 ∪ 𝐺) ∖ {∅}) ∧ (𝐹 ∪ 𝐺) ∈ V ∧ dom (𝐹 ∪ 𝐺) ⊆ (𝐴...𝐷))
75		isstructr 13419	. 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 430	1 ⊢ (𝐹 ∪ 𝐺) Struct ⟨𝐴, 𝐷⟩

This proof depends on syntax axioms:   ∧ wa 104   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ∖ cdif 3217   ∪ cun 3218   ∩ cin 3219   ⊆ wss 3220  ∅c0 3520  {csn 3709  ⟨cop 3712   class class class wbr 4130  dom cdm 4774  Fun wfun 5371  ‘cfv 5377  (class class class)co 6085   < clt 8361   ≤ cle 8362  ℕcn 9307  ℤcz 9649  ℤ_≥cuz 9931  ...cfz 10422   Struct cstr 13400

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-addcom 8280  ax-addass 8282  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-0id 8288  ax-rnegex 8289  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-ltadd 8296

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-inn 9308  df-z 9650  df-uz 9932  df-fz 10423  df-struct 13406

This theorem is used by:  cnfldstr  14979