Theorem om2uzlt 4654

Description: Less-than relation for G (see om2uz0 4651).

Hypotheses

Ref	Expression
om2uz.1	⊢ C ∈ ℤ
om2uz.2	⊢ G = (rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω)

Assertion

Ref	Expression
om2uzlt	⊢ ((A ∈ ω ∧ B ∈ ω) → (A ∈ B → (G ‘A) < (G ‘B)))

Distinct variable group(s):   x,y,C

Proof of Theorem om2uzlt

Step	Hyp	Ref	Expression
1		eleq2 1150	. . . . . 6 ⊢ (v = ∅ → (A ∈ v ↔ A ∈ ∅))
2		fveq2 2832	. . . . . . 7 ⊢ (v = ∅ → (G ‘v) = (G ‘∅))
3	2	breq2d 2072	. . . . . 6 ⊢ (v = ∅ → ((G ‘A) < (G ‘v) ↔ (G ‘A) < (G ‘∅)))
4	1, 3	imbi12d 474	. . . . 5 ⊢ (v = ∅ → ((A ∈ v → (G ‘A) < (G ‘v)) ↔ (A ∈ ∅ → (G ‘A) < (G ‘∅))))
5	4	imbi2d 464	. . . 4 ⊢ (v = ∅ → ((A ∈ ω → (A ∈ v → (G ‘A) < (G ‘v))) ↔ (A ∈ ω → (A ∈ ∅ → (G ‘A) < (G ‘∅)))))
6		eleq2 1150	. . . . . 6 ⊢ (v = w → (A ∈ v ↔ A ∈ w))
7		fveq2 2832	. . . . . . 7 ⊢ (v = w → (G ‘v) = (G ‘w))
8	7	breq2d 2072	. . . . . 6 ⊢ (v = w → ((G ‘A) < (G ‘v) ↔ (G ‘A) < (G ‘w)))
9	6, 8	imbi12d 474	. . . . 5 ⊢ (v = w → ((A ∈ v → (G ‘A) < (G ‘v)) ↔ (A ∈ w → (G ‘A) < (G ‘w))))
10	9	imbi2d 464	. . . 4 ⊢ (v = w → ((A ∈ ω → (A ∈ v → (G ‘A) < (G ‘v))) ↔ (A ∈ ω → (A ∈ w → (G ‘A) < (G ‘w)))))
11		eleq2 1150	. . . . . 6 ⊢ (v = suc w → (A ∈ v ↔ A ∈ suc w))
12		fveq2 2832	. . . . . . 7 ⊢ (v = suc w → (G ‘v) = (G ‘suc w))
13	12	breq2d 2072	. . . . . 6 ⊢ (v = suc w → ((G ‘A) < (G ‘v) ↔ (G ‘A) < (G ‘suc w)))
14	11, 13	imbi12d 474	. . . . 5 ⊢ (v = suc w → ((A ∈ v → (G ‘A) < (G ‘v)) ↔ (A ∈ suc w → (G ‘A) < (G ‘suc w))))
15	14	imbi2d 464	. . . 4 ⊢ (v = suc w → ((A ∈ ω → (A ∈ v → (G ‘A) < (G ‘v))) ↔ (A ∈ ω → (A ∈ suc w → (G ‘A) < (G ‘suc w)))))
16		eleq2 1150	. . . . . 6 ⊢ (v = B → (A ∈ v ↔ A ∈ B))
17		fveq2 2832	. . . . . . 7 ⊢ (v = B → (G ‘v) = (G ‘B))
18	17	breq2d 2072	. . . . . 6 ⊢ (v = B → ((G ‘A) < (G ‘v) ↔ (G ‘A) < (G ‘B)))
19	16, 18	imbi12d 474	. . . . 5 ⊢ (v = B → ((A ∈ v → (G ‘A) < (G ‘v)) ↔ (A ∈ B → (G ‘A) < (G ‘B))))
20	19	imbi2d 464	. . . 4 ⊢ (v = B → ((A ∈ ω → (A ∈ v → (G ‘A) < (G ‘v))) ↔ (A ∈ ω → (A ∈ B → (G ‘A) < (G ‘B)))))
21		noel 1711	. . . . . 6 ⊢ ¬ A ∈ ∅
22	21	pm2.21i 73	. . . . 5 ⊢ (A ∈ ∅ → (G ‘A) < (G ‘∅))
23	22	a1i 7	. . . 4 ⊢ (A ∈ ω → (A ∈ ∅ → (G ‘A) < (G ‘∅)))
24		elsuc2g 2291	. . . . . . . . . . 11 ⊢ (w ∈ ω → (A ∈ suc w ↔ (A ∈ w ∨ A = w)))
25	24	bicomd 399	. . . . . . . . . 10 ⊢ (w ∈ ω → ((A ∈ w ∨ A = w) ↔ A ∈ suc w))
26	25	adantl 305	. . . . . . . . 9 ⊢ ((A ∈ ω ∧ w ∈ ω) → ((A ∈ w ∨ A = w) ↔ A ∈ suc w))
27		leloet 4284	. . . . . . . . . . 11 ⊢ (((G ‘A) ∈ ℝ ∧ (G ‘w) ∈ ℝ) → ((G ‘A) ≤ (G ‘w) ↔ ((G ‘A) < (G ‘w) ∨ (G ‘A) = (G ‘w))))
28		om2uz.1	. . . . . . . . . . . . 13 ⊢ C ∈ ℤ
29		om2uz.2	. . . . . . . . . . . . 13 ⊢ G = (rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω)
30	28, 29	om2uzuz 4653	. . . . . . . . . . . 12 ⊢ (A ∈ ω → (G ‘A) ∈ {z ∈ ℤ∣C ≤ z})
31		ssrab 1556	. . . . . . . . . . . . 13 ⊢ {z ∈ ℤ∣C ≤ z} ⊆ ℤ
32	31	sseli 1504	. . . . . . . . . . . 12 ⊢ ((G ‘A) ∈ {z ∈ ℤ∣C ≤ z} → (G ‘A) ∈ ℤ)
33		zret 4567	. . . . . . . . . . . 12 ⊢ ((G ‘A) ∈ ℤ → (G ‘A) ∈ ℝ)
34	30, 32, 33	3syl 21	. . . . . . . . . . 11 ⊢ (A ∈ ω → (G ‘A) ∈ ℝ)
35	28, 29	om2uzuz 4653	. . . . . . . . . . . 12 ⊢ (w ∈ ω → (G ‘w) ∈ {z ∈ ℤ∣C ≤ z})
36	31	sseli 1504	. . . . . . . . . . . 12 ⊢ ((G ‘w) ∈ {z ∈ ℤ∣C ≤ z} → (G ‘w) ∈ ℤ)
37		zret 4567	. . . . . . . . . . . 12 ⊢ ((G ‘w) ∈ ℤ → (G ‘w) ∈ ℝ)
38	35, 36, 37	3syl 21	. . . . . . . . . . 11 ⊢ (w ∈ ω → (G ‘w) ∈ ℝ)
39	27, 34, 38	syl2an 349	. . . . . . . . . 10 ⊢ ((A ∈ ω ∧ w ∈ ω) → ((G ‘A) ≤ (G ‘w) ↔ ((G ‘A) < (G ‘w) ∨ (G ‘A) = (G ‘w))))
40		zleltp1t 4598	. . . . . . . . . . . . 13 ⊢ (((G ‘A) ∈ ℤ ∧ (G ‘w) ∈ ℤ) → ((G ‘A) ≤ (G ‘w) ↔ (G ‘A) < ((G ‘w) + 1)))
41	40, 32, 36	syl2an 349	. . . . . . . . . . . 12 ⊢ (((G ‘A) ∈ {z ∈ ℤ∣C ≤ z} ∧ (G ‘w) ∈ {z ∈ ℤ∣C ≤ z}) → ((G ‘A) ≤ (G ‘w) ↔ (G ‘A) < ((G ‘w) + 1)))
42	41, 30, 35	syl2an 349	. . . . . . . . . . 11 ⊢ ((A ∈ ω ∧ w ∈ ω) → ((G ‘A) ≤ (G ‘w) ↔ (G ‘A) < ((G ‘w) + 1)))
43	28, 29	om2uzsuc 4652	. . . . . . . . . . . . 13 ⊢ (w ∈ ω → (G ‘suc w) = ((G ‘w) + 1))
44	43	breq2d 2072	. . . . . . . . . . . 12 ⊢ (w ∈ ω → ((G ‘A) < (G ‘suc w) ↔ (G ‘A) < ((G ‘w) + 1)))
45	44	adantl 305	. . . . . . . . . . 11 ⊢ ((A ∈ ω ∧ w ∈ ω) → ((G ‘A) < (G ‘suc w) ↔ (G ‘A) < ((G ‘w) + 1)))
46	42, 45	bitr4d 409	. . . . . . . . . 10 ⊢ ((A ∈ ω ∧ w ∈ ω) → ((G ‘A) ≤ (G ‘w) ↔ (G ‘A) < (G ‘suc w)))
47	39, 46	bitr3d 408	. . . . . . . . 9 ⊢ ((A ∈ ω ∧ w ∈ ω) → (((G ‘A) < (G ‘w) ∨ (G ‘A) = (G ‘w)) ↔ (G ‘A) < (G ‘suc w)))
48	26, 47	imbi12d 474	. . . . . . . 8 ⊢ ((A ∈ ω ∧ w ∈ ω) → (((A ∈ w ∨ A = w) → ((G ‘A) < (G ‘w) ∨ (G ‘A) = (G ‘w))) ↔ (A ∈ suc w → (G ‘A) < (G ‘suc w))))
49		id 9	. . . . . . . . 9 ⊢ ((A ∈ w → (G ‘A) < (G ‘w)) → (A ∈ w → (G ‘A) < (G ‘w)))
50		fveq2 2832	. . . . . . . . . 10 ⊢ (A = w → (G ‘A) = (G ‘w))
51	50	a1i 7	. . . . . . . . 9 ⊢ ((A ∈ w → (G ‘A) < (G ‘w)) → (A = w → (G ‘A) = (G ‘w)))
52	49, 51	orim12d 436	. . . . . . . 8 ⊢ ((A ∈ w → (G ‘A) < (G ‘w)) → ((A ∈ w ∨ A = w) → ((G ‘A) < (G ‘w) ∨ (G ‘A) = (G ‘w))))
53	48, 52	syl5bi 183	. . . . . . 7 ⊢ ((A ∈ ω ∧ w ∈ ω) → ((A ∈ w → (G ‘A) < (G ‘w)) → (A ∈ suc w → (G ‘A) < (G ‘suc w))))
54	53	exp 291	. . . . . 6 ⊢ (A ∈ ω → (w ∈ ω → ((A ∈ w → (G ‘A) < (G ‘w)) → (A ∈ suc w → (G ‘A) < (G ‘suc w)))))
55	54	com12 13	. . . . 5 ⊢ (w ∈ ω → (A ∈ ω → ((A ∈ w → (G ‘A) < (G ‘w)) → (A ∈ suc w → (G ‘A) < (G ‘suc w)))))
56	55	a2d 15	. . . 4 ⊢ (w ∈ ω → ((A ∈ ω → (A ∈ w → (G ‘A) < (G ‘w))) → (A ∈ ω → (A ∈ suc w → (G ‘A) < (G ‘suc w)))))
57	5, 10, 15, 20, 23, 56	finds 2397	. . 3 ⊢ (B ∈ ω → (A ∈ ω → (A ∈ B → (G ‘A) < (G ‘B))))
58	57	com12 13	. 2 ⊢ (A ∈ ω → (B ∈ ω → (A ∈ B → (G ‘A) < (G ‘B))))
59	58	imp 277	1 ⊢ ((A ∈ ω ∧ B ∈ ω) → (A ∈ B → (G ‘A) < (G ‘B)))

Syntax hints:   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   = weq 797   = wceq 1091   ∈ wcel 1092  {crab 1204  ∅c0 1707   class class class wbr 2054  {copab 2055  suc csuc 2201  ωcom 2372   ↾ cres 2412   ‘cfv 2422  reccrdg 2969  (class class class)co 3001  ℝcr 4027  1c1 4029   + caddc 4031   < clt 4033   ≤ cle 4092  ℤcz 4095

This theorem is referenced by:  om2uzf1o 4656

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-le 4277  df-n 4423  df-n0 4535  df-z 4564

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