Theorem strleun 11732

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 11657	. . . . . 6 ⊢ (𝐹 Struct ⟨𝐴, 𝐵⟩ → ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐴 ≤ 𝐵) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (𝐴...𝐵)))
3	1, 2	ax-mp 7	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐴 ≤ 𝐵) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (𝐴...𝐵))
4	3	simp1i 955	. . . 4 ⊢ (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐴 ≤ 𝐵)
5	4	simp1i 955	. . 3 ⊢ 𝐴 ∈ ℕ
6		strleun.g	. . . . . 6 ⊢ 𝐺 Struct ⟨𝐶, 𝐷⟩
7		isstructim 11657	. . . . . 6 ⊢ (𝐺 Struct ⟨𝐶, 𝐷⟩ → ((𝐶 ∈ ℕ ∧ 𝐷 ∈ ℕ ∧ 𝐶 ≤ 𝐷) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (𝐶...𝐷)))
8	6, 7	ax-mp 7	. . . . 5 ⊢ ((𝐶 ∈ ℕ ∧ 𝐷 ∈ ℕ ∧ 𝐶 ≤ 𝐷) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (𝐶...𝐷))
9	8	simp1i 955	. . . 4 ⊢ (𝐶 ∈ ℕ ∧ 𝐷 ∈ ℕ ∧ 𝐶 ≤ 𝐷)
10	9	simp2i 956	. . 3 ⊢ 𝐷 ∈ ℕ
11	4	simp3i 957	. . . . 5 ⊢ 𝐴 ≤ 𝐵
12	4	simp2i 956	. . . . . . 7 ⊢ 𝐵 ∈ ℕ
13	12	nnrei 8529	. . . . . 6 ⊢ 𝐵 ∈ ℝ
14	9	simp1i 955	. . . . . . 7 ⊢ 𝐶 ∈ ℕ
15	14	nnrei 8529	. . . . . 6 ⊢ 𝐶 ∈ ℝ
16		strleun.l	. . . . . 6 ⊢ 𝐵 < 𝐶
17	13, 15, 16	ltleii 7684	. . . . 5 ⊢ 𝐵 ≤ 𝐶
18	5	nnrei 8529	. . . . . 6 ⊢ 𝐴 ∈ ℝ
19	18, 13, 15	letri 7689	. . . . 5 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)
20	11, 17, 19	mp2an 418	. . . 4 ⊢ 𝐴 ≤ 𝐶
21	9	simp3i 957	. . . 4 ⊢ 𝐶 ≤ 𝐷
22	10	nnrei 8529	. . . . 5 ⊢ 𝐷 ∈ ℝ
23	18, 15, 22	letri 7689	. . . 4 ⊢ ((𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐷) → 𝐴 ≤ 𝐷)
24	20, 21, 23	mp2an 418	. . 3 ⊢ 𝐴 ≤ 𝐷
25	5, 10, 24	3pm3.2i 1124	. 2 ⊢ (𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ∧ 𝐴 ≤ 𝐷)
26	3	simp2i 956	. . . . . 6 ⊢ Fun (𝐹 ∖ {∅})
27	8	simp2i 956	. . . . . 6 ⊢ Fun (𝐺 ∖ {∅})
28	26, 27	pm3.2i 267	. . . . 5 ⊢ (Fun (𝐹 ∖ {∅}) ∧ Fun (𝐺 ∖ {∅}))
29		difss 3141	. . . . . . . . 9 ⊢ (𝐹 ∖ {∅}) ⊆ 𝐹
30		dmss 4666	. . . . . . . . 9 ⊢ ((𝐹 ∖ {∅}) ⊆ 𝐹 → dom (𝐹 ∖ {∅}) ⊆ dom 𝐹)
31	29, 30	ax-mp 7	. . . . . . . 8 ⊢ dom (𝐹 ∖ {∅}) ⊆ dom 𝐹
32	3	simp3i 957	. . . . . . . 8 ⊢ dom 𝐹 ⊆ (𝐴...𝐵)
33	31, 32	sstri 3048	. . . . . . 7 ⊢ dom (𝐹 ∖ {∅}) ⊆ (𝐴...𝐵)
34		difss 3141	. . . . . . . . 9 ⊢ (𝐺 ∖ {∅}) ⊆ 𝐺
35		dmss 4666	. . . . . . . . 9 ⊢ ((𝐺 ∖ {∅}) ⊆ 𝐺 → dom (𝐺 ∖ {∅}) ⊆ dom 𝐺)
36	34, 35	ax-mp 7	. . . . . . . 8 ⊢ dom (𝐺 ∖ {∅}) ⊆ dom 𝐺
37	8	simp3i 957	. . . . . . . 8 ⊢ dom 𝐺 ⊆ (𝐶...𝐷)
38	36, 37	sstri 3048	. . . . . . 7 ⊢ dom (𝐺 ∖ {∅}) ⊆ (𝐶...𝐷)
39		ss2in 3243	. . . . . . 7 ⊢ ((dom (𝐹 ∖ {∅}) ⊆ (𝐴...𝐵) ∧ dom (𝐺 ∖ {∅}) ⊆ (𝐶...𝐷)) → (dom (𝐹 ∖ {∅}) ∩ dom (𝐺 ∖ {∅})) ⊆ ((𝐴...𝐵) ∩ (𝐶...𝐷)))
40	33, 38, 39	mp2an 418	. . . . . 6 ⊢ (dom (𝐹 ∖ {∅}) ∩ dom (𝐺 ∖ {∅})) ⊆ ((𝐴...𝐵) ∩ (𝐶...𝐷))
41		fzdisj 9615	. . . . . . 7 ⊢ (𝐵 < 𝐶 → ((𝐴...𝐵) ∩ (𝐶...𝐷)) = ∅)
42	16, 41	ax-mp 7	. . . . . 6 ⊢ ((𝐴...𝐵) ∩ (𝐶...𝐷)) = ∅
43		sseq0 3343	. . . . . 6 ⊢ (((dom (𝐹 ∖ {∅}) ∩ dom (𝐺 ∖ {∅})) ⊆ ((𝐴...𝐵) ∩ (𝐶...𝐷)) ∧ ((𝐴...𝐵) ∩ (𝐶...𝐷)) = ∅) → (dom (𝐹 ∖ {∅}) ∩ dom (𝐺 ∖ {∅})) = ∅)
44	40, 42, 43	mp2an 418	. . . . 5 ⊢ (dom (𝐹 ∖ {∅}) ∩ dom (𝐺 ∖ {∅})) = ∅
45		funun 5092	. . . . 5 ⊢ (((Fun (𝐹 ∖ {∅}) ∧ Fun (𝐺 ∖ {∅})) ∧ (dom (𝐹 ∖ {∅}) ∩ dom (𝐺 ∖ {∅})) = ∅) → Fun ((𝐹 ∖ {∅}) ∪ (𝐺 ∖ {∅})))
46	28, 44, 45	mp2an 418	. . . 4 ⊢ Fun ((𝐹 ∖ {∅}) ∪ (𝐺 ∖ {∅}))
47		difundir 3268	. . . . 5 ⊢ ((𝐹 ∪ 𝐺) ∖ {∅}) = ((𝐹 ∖ {∅}) ∪ (𝐺 ∖ {∅}))
48	47	funeqi 5070	. . . 4 ⊢ (Fun ((𝐹 ∪ 𝐺) ∖ {∅}) ↔ Fun ((𝐹 ∖ {∅}) ∪ (𝐺 ∖ {∅})))
49	46, 48	mpbir 145	. . 3 ⊢ Fun ((𝐹 ∪ 𝐺) ∖ {∅})
50		structex 11655	. . . . 5 ⊢ (𝐹 Struct ⟨𝐴, 𝐵⟩ → 𝐹 ∈ V)
51	1, 50	ax-mp 7	. . . 4 ⊢ 𝐹 ∈ V
52		structex 11655	. . . . 5 ⊢ (𝐺 Struct ⟨𝐶, 𝐷⟩ → 𝐺 ∈ V)
53	6, 52	ax-mp 7	. . . 4 ⊢ 𝐺 ∈ V
54	51, 53	unex 4291	. . 3 ⊢ (𝐹 ∪ 𝐺) ∈ V
55		dmun 4674	. . . 4 ⊢ dom (𝐹 ∪ 𝐺) = (dom 𝐹 ∪ dom 𝐺)
56	12	nnzi 8869	. . . . . . . 8 ⊢ 𝐵 ∈ ℤ
57	10	nnzi 8869	. . . . . . . 8 ⊢ 𝐷 ∈ ℤ
58	13, 15, 22	letri 7689	. . . . . . . . 9 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐷) → 𝐵 ≤ 𝐷)
59	17, 21, 58	mp2an 418	. . . . . . . 8 ⊢ 𝐵 ≤ 𝐷
60		eluz2 9124	. . . . . . . 8 ⊢ (𝐷 ∈ (ℤ≥‘𝐵) ↔ (𝐵 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐵 ≤ 𝐷))
61	56, 57, 59, 60	mpbir3an 1128	. . . . . . 7 ⊢ 𝐷 ∈ (ℤ≥‘𝐵)
62		fzss2 9627	. . . . . . 7 ⊢ (𝐷 ∈ (ℤ≥‘𝐵) → (𝐴...𝐵) ⊆ (𝐴...𝐷))
63	61, 62	ax-mp 7	. . . . . 6 ⊢ (𝐴...𝐵) ⊆ (𝐴...𝐷)
64	32, 63	sstri 3048	. . . . 5 ⊢ dom 𝐹 ⊆ (𝐴...𝐷)
65	5	nnzi 8869	. . . . . . . 8 ⊢ 𝐴 ∈ ℤ
66	14	nnzi 8869	. . . . . . . 8 ⊢ 𝐶 ∈ ℤ
67		eluz2 9124	. . . . . . . 8 ⊢ (𝐶 ∈ (ℤ≥‘𝐴) ↔ (𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐴 ≤ 𝐶))
68	65, 66, 20, 67	mpbir3an 1128	. . . . . . 7 ⊢ 𝐶 ∈ (ℤ≥‘𝐴)
69		fzss1 9626	. . . . . . 7 ⊢ (𝐶 ∈ (ℤ≥‘𝐴) → (𝐶...𝐷) ⊆ (𝐴...𝐷))
70	68, 69	ax-mp 7	. . . . . 6 ⊢ (𝐶...𝐷) ⊆ (𝐴...𝐷)
71	37, 70	sstri 3048	. . . . 5 ⊢ dom 𝐺 ⊆ (𝐴...𝐷)
72	64, 71	unssi 3190	. . . 4 ⊢ (dom 𝐹 ∪ dom 𝐺) ⊆ (𝐴...𝐷)
73	55, 72	eqsstri 3071	. . 3 ⊢ dom (𝐹 ∪ 𝐺) ⊆ (𝐴...𝐷)
74	49, 54, 73	3pm3.2i 1124	. 2 ⊢ (Fun ((𝐹 ∪ 𝐺) ∖ {∅}) ∧ (𝐹 ∪ 𝐺) ∈ V ∧ dom (𝐹 ∪ 𝐺) ⊆ (𝐴...𝐷))
75		isstructr 11658	. 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 418	1 ⊢ (𝐹 ∪ 𝐺) Struct ⟨𝐴, 𝐷⟩

Syntax hints:   ∧ wa 103   ∧ w3a 927   = wceq 1296   ∈ wcel 1445  Vcvv 2633   ∖ cdif 3010   ∪ cun 3011   ∩ cin 3012   ⊆ wss 3013  ∅c0 3302  {csn 3466  ⟨cop 3469   class class class wbr 3867  dom cdm 4467  Fun wfun 5043  ‘cfv 5049  (class class class)co 5690   < clt 7619   ≤ cle 7620  ℕcn 8520  ℤcz 8848  ℤ_≥cuz 9118  ...cfz 9573   Struct cstr 11639

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-addcom 7542  ax-addass 7544  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-0id 7550  ax-rnegex 7551  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-ltadd 7558

This theorem depends on definitions:  df-bi 116  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-inn 8521  df-z 8849  df-uz 9119  df-fz 9574  df-struct 11645

This theorem is referenced by: (None)