Theorem sqr2irrlem2 4778

Description: Lemma for irrationality of square root of 2. Eliminates hypotheses with weak deduction theorem.

Assertion

Ref	Expression
sqr2irrlem2	⊢ ((A ∈ ℕ ∧ B ∈ ℕ) → ((A↑2) = (2 · (B↑2)) → ((B < A ∧ (A / 2) ∈ ℕ) ∧ (B↑2) = (2 · ((A / 2)↑2)))))

Proof of Theorem sqr2irrlem2

Step	Hyp	Ref	Expression
1		opreq1 3006	. . . 4 ⊢ (A = if(A ∈ ℕ, A, 2) → (A↑2) = (if(A ∈ ℕ, A, 2)↑2))
2	1	cleq1d 1109	. . 3 ⊢ (A = if(A ∈ ℕ, A, 2) → ((A↑2) = (2 · (B↑2)) ↔ (if(A ∈ ℕ, A, 2)↑2) = (2 · (B↑2))))
3		breq2 2066	. . . . 5 ⊢ (A = if(A ∈ ℕ, A, 2) → (B < A ↔ B < if(A ∈ ℕ, A, 2)))
4		opreq1 3006	. . . . . 6 ⊢ (A = if(A ∈ ℕ, A, 2) → (A / 2) = (if(A ∈ ℕ, A, 2) / 2))
5	4	eleq1d 1155	. . . . 5 ⊢ (A = if(A ∈ ℕ, A, 2) → ((A / 2) ∈ ℕ ↔ (if(A ∈ ℕ, A, 2) / 2) ∈ ℕ))
6	3, 5	anbi12d 476	. . . 4 ⊢ (A = if(A ∈ ℕ, A, 2) → ((B < A ∧ (A / 2) ∈ ℕ) ↔ (B < if(A ∈ ℕ, A, 2) ∧ (if(A ∈ ℕ, A, 2) / 2) ∈ ℕ)))
7	4	opreq1d 3012	. . . . . 6 ⊢ (A = if(A ∈ ℕ, A, 2) → ((A / 2)↑2) = ((if(A ∈ ℕ, A, 2) / 2)↑2))
8	7	opreq2d 3013	. . . . 5 ⊢ (A = if(A ∈ ℕ, A, 2) → (2 · ((A / 2)↑2)) = (2 · ((if(A ∈ ℕ, A, 2) / 2)↑2)))
9	8	cleq2d 1112	. . . 4 ⊢ (A = if(A ∈ ℕ, A, 2) → ((B↑2) = (2 · ((A / 2)↑2)) ↔ (B↑2) = (2 · ((if(A ∈ ℕ, A, 2) / 2)↑2))))
10	6, 9	anbi12d 476	. . 3 ⊢ (A = if(A ∈ ℕ, A, 2) → (((B < A ∧ (A / 2) ∈ ℕ) ∧ (B↑2) = (2 · ((A / 2)↑2))) ↔ ((B < if(A ∈ ℕ, A, 2) ∧ (if(A ∈ ℕ, A, 2) / 2) ∈ ℕ) ∧ (B↑2) = (2 · ((if(A ∈ ℕ, A, 2) / 2)↑2)))))
11	2, 10	imbi12d 474	. 2 ⊢ (A = if(A ∈ ℕ, A, 2) → (((A↑2) = (2 · (B↑2)) → ((B < A ∧ (A / 2) ∈ ℕ) ∧ (B↑2) = (2 · ((A / 2)↑2)))) ↔ ((if(A ∈ ℕ, A, 2)↑2) = (2 · (B↑2)) → ((B < if(A ∈ ℕ, A, 2) ∧ (if(A ∈ ℕ, A, 2) / 2) ∈ ℕ) ∧ (B↑2) = (2 · ((if(A ∈ ℕ, A, 2) / 2)↑2))))))
12		opreq1 3006	. . . . 5 ⊢ (B = if(B ∈ ℕ, B, 2) → (B↑2) = (if(B ∈ ℕ, B, 2)↑2))
13	12	opreq2d 3013	. . . 4 ⊢ (B = if(B ∈ ℕ, B, 2) → (2 · (B↑2)) = (2 · (if(B ∈ ℕ, B, 2)↑2)))
14	13	cleq2d 1112	. . 3 ⊢ (B = if(B ∈ ℕ, B, 2) → ((if(A ∈ ℕ, A, 2)↑2) = (2 · (B↑2)) ↔ (if(A ∈ ℕ, A, 2)↑2) = (2 · (if(B ∈ ℕ, B, 2)↑2))))
15		breq1 2065	. . . . 5 ⊢ (B = if(B ∈ ℕ, B, 2) → (B < if(A ∈ ℕ, A, 2) ↔ if(B ∈ ℕ, B, 2) < if(A ∈ ℕ, A, 2)))
16	15	anbi1d 469	. . . 4 ⊢ (B = if(B ∈ ℕ, B, 2) → ((B < if(A ∈ ℕ, A, 2) ∧ (if(A ∈ ℕ, A, 2) / 2) ∈ ℕ) ↔ (if(B ∈ ℕ, B, 2) < if(A ∈ ℕ, A, 2) ∧ (if(A ∈ ℕ, A, 2) / 2) ∈ ℕ)))
17	12	cleq1d 1109	. . . 4 ⊢ (B = if(B ∈ ℕ, B, 2) → ((B↑2) = (2 · ((if(A ∈ ℕ, A, 2) / 2)↑2)) ↔ (if(B ∈ ℕ, B, 2)↑2) = (2 · ((if(A ∈ ℕ, A, 2) / 2)↑2))))
18	16, 17	anbi12d 476	. . 3 ⊢ (B = if(B ∈ ℕ, B, 2) → (((B < if(A ∈ ℕ, A, 2) ∧ (if(A ∈ ℕ, A, 2) / 2) ∈ ℕ) ∧ (B↑2) = (2 · ((if(A ∈ ℕ, A, 2) / 2)↑2))) ↔ ((if(B ∈ ℕ, B, 2) < if(A ∈ ℕ, A, 2) ∧ (if(A ∈ ℕ, A, 2) / 2) ∈ ℕ) ∧ (if(B ∈ ℕ, B, 2)↑2) = (2 · ((if(A ∈ ℕ, A, 2) / 2)↑2)))))
19	14, 18	imbi12d 474	. 2 ⊢ (B = if(B ∈ ℕ, B, 2) → (((if(A ∈ ℕ, A, 2)↑2) = (2 · (B↑2)) → ((B < if(A ∈ ℕ, A, 2) ∧ (if(A ∈ ℕ, A, 2) / 2) ∈ ℕ) ∧ (B↑2) = (2 · ((if(A ∈ ℕ, A, 2) / 2)↑2)))) ↔ ((if(A ∈ ℕ, A, 2)↑2) = (2 · (if(B ∈ ℕ, B, 2)↑2)) → ((if(B ∈ ℕ, B, 2) < if(A ∈ ℕ, A, 2) ∧ (if(A ∈ ℕ, A, 2) / 2) ∈ ℕ) ∧ (if(B ∈ ℕ, B, 2)↑2) = (2 · ((if(A ∈ ℕ, A, 2) / 2)↑2))))))
20		2nn 4487	. . . 4 ⊢ 2 ∈ ℕ
21	20	elimel 1793	. . 3 ⊢ if(A ∈ ℕ, A, 2) ∈ ℕ
22	20	elimel 1793	. . 3 ⊢ if(B ∈ ℕ, B, 2) ∈ ℕ
23	21, 22	sqr2irrlem1 4777	. 2 ⊢ ((if(A ∈ ℕ, A, 2)↑2) = (2 · (if(B ∈ ℕ, B, 2)↑2)) → ((if(B ∈ ℕ, B, 2) < if(A ∈ ℕ, A, 2) ∧ (if(A ∈ ℕ, A, 2) / 2) ∈ ℕ) ∧ (if(B ∈ ℕ, B, 2)↑2) = (2 · ((if(A ∈ ℕ, A, 2) / 2)↑2))))
24	11, 19, 23	dedth2h 1787	1 ⊢ ((A ∈ ℕ ∧ B ∈ ℕ) → ((A↑2) = (2 · (B↑2)) → ((B < A ∧ (A / 2) ∈ ℕ) ∧ (B↑2) = (2 · ((A / 2)↑2)))))

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ifcif 1776   class class class wbr 2054  (class class class)co 3001   · cmulc 4032   < clt 4033   / cdiv 4091  ℕcn 4093  2c2 4454  ↑cexp 4675

This theorem is referenced by:  sqr2irrlem3 4779

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-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1st 3087  df-2nd 3088  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-div 4216  df-le 4277  df-n 4423  df-2 4462  df-n0 4535  df-z 4564  df-seq 4661  df-exp 4676

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