Theorem 4sqlem2 13146

Description: Lemma for 4sq 13167. Change bound variables in 𝑆. (Contributed by Mario Carneiro, 14-Jul-2014.)

Hypothesis

Ref	Expression
4sq.1	⊢ 𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}

Assertion

Ref	Expression
4sqlem2	⊢ (𝐴 ∈ 𝑆 ↔ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ ∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ 𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))))

Distinct variable groups:   𝑎,𝑏,𝑐,𝑑,𝑛,𝑤,𝑥,𝑦,𝑧   𝐴,𝑎,𝑏,𝑐,𝑑,𝑛   𝑆,𝑎,𝑏,𝑐,𝑑,𝑛

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧,𝑤)   𝑆(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem 4sqlem2

Step	Hyp	Ref	Expression
1		4sq.1	. . 3 ⊢ 𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}
2	1	eleq2i 2305	. 2 ⊢ (𝐴 ∈ 𝑆 ↔ 𝐴 ∈ {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))})
3		zsqcl2 11032	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℤ → (𝑎↑2) ∈ ℕ0)
4	3	ad2antrr 492	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ)) → (𝑎↑2) ∈ ℕ0)
5		zsqcl2 11032	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℤ → (𝑏↑2) ∈ ℕ0)
6	5	ad2antlr 493	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ)) → (𝑏↑2) ∈ ℕ0)
7	4, 6	nn0addcld 9603	. . . . . . . 8 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ)) → ((𝑎↑2) + (𝑏↑2)) ∈ ℕ0)
8		zsqcl2 11032	. . . . . . . . . 10 ⊢ (𝑐 ∈ ℤ → (𝑐↑2) ∈ ℕ0)
9	8	ad2antrl 494	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ)) → (𝑐↑2) ∈ ℕ0)
10		zsqcl2 11032	. . . . . . . . . 10 ⊢ (𝑑 ∈ ℤ → (𝑑↑2) ∈ ℕ0)
11	10	ad2antll 495	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ)) → (𝑑↑2) ∈ ℕ0)
12	9, 11	nn0addcld 9603	. . . . . . . 8 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ)) → ((𝑐↑2) + (𝑑↑2)) ∈ ℕ0)
13	7, 12	nn0addcld 9603	. . . . . . 7 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ)) → (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))) ∈ ℕ0)
14		eleq1 2301	. . . . . . 7 ⊢ (𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))) → (𝐴 ∈ ℕ0 ↔ (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))) ∈ ℕ0))
15	13, 14	syl5ibrcom 157	. . . . . 6 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ)) → (𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))) → 𝐴 ∈ ℕ0))
16		elex 2833	. . . . . 6 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ V)
17	15, 16	syl6 33	. . . . 5 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ)) → (𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))) → 𝐴 ∈ V))
18	17	rexlimdvva 2676	. . . 4 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → (∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ 𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))) → 𝐴 ∈ V))
19	18	rexlimivv 2674	. . 3 ⊢ (∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ ∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ 𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))) → 𝐴 ∈ V)
20		oveq1 6082	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝑥↑2) = (𝑎↑2))
21	20	oveq1d 6090	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ((𝑥↑2) + (𝑦↑2)) = ((𝑎↑2) + (𝑦↑2)))
22	21	oveq1d 6090	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) = (((𝑎↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
23	22	eqeq2d 2250	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ 𝑛 = (((𝑎↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))))
24	23	2rexbidv 2575	. . . . 5 ⊢ (𝑥 = 𝑎 → (∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑎↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))))
25		oveq1 6082	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → (𝑦↑2) = (𝑏↑2))
26	25	oveq2d 6091	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → ((𝑎↑2) + (𝑦↑2)) = ((𝑎↑2) + (𝑏↑2)))
27	26	oveq1d 6090	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (((𝑎↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) = (((𝑎↑2) + (𝑏↑2)) + ((𝑧↑2) + (𝑤↑2))))
28	27	eqeq2d 2250	. . . . . 6 ⊢ (𝑦 = 𝑏 → (𝑛 = (((𝑎↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ 𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑧↑2) + (𝑤↑2)))))
29	28	2rexbidv 2575	. . . . 5 ⊢ (𝑦 = 𝑏 → (∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑎↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑧↑2) + (𝑤↑2)))))
30	24, 29	cbvrex2vw 2798	. . . 4 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑧↑2) + (𝑤↑2))))
31		oveq1 6082	. . . . . . . . . 10 ⊢ (𝑧 = 𝑐 → (𝑧↑2) = (𝑐↑2))
32	31	oveq1d 6090	. . . . . . . . 9 ⊢ (𝑧 = 𝑐 → ((𝑧↑2) + (𝑤↑2)) = ((𝑐↑2) + (𝑤↑2)))
33	32	oveq2d 6091	. . . . . . . 8 ⊢ (𝑧 = 𝑐 → (((𝑎↑2) + (𝑏↑2)) + ((𝑧↑2) + (𝑤↑2))) = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑤↑2))))
34	33	eqeq2d 2250	. . . . . . 7 ⊢ (𝑧 = 𝑐 → (𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ 𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑤↑2)))))
35		oveq1 6082	. . . . . . . . . 10 ⊢ (𝑤 = 𝑑 → (𝑤↑2) = (𝑑↑2))
36	35	oveq2d 6091	. . . . . . . . 9 ⊢ (𝑤 = 𝑑 → ((𝑐↑2) + (𝑤↑2)) = ((𝑐↑2) + (𝑑↑2)))
37	36	oveq2d 6091	. . . . . . . 8 ⊢ (𝑤 = 𝑑 → (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑤↑2))) = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))))
38	37	eqeq2d 2250	. . . . . . 7 ⊢ (𝑤 = 𝑑 → (𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑤↑2))) ↔ 𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2)))))
39	34, 38	cbvrex2vw 2798	. . . . . 6 ⊢ (∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ ∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ 𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))))
40		eqeq1 2245	. . . . . . 7 ⊢ (𝑛 = 𝐴 → (𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))) ↔ 𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2)))))
41	40	2rexbidv 2575	. . . . . 6 ⊢ (𝑛 = 𝐴 → (∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ 𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))) ↔ ∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ 𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2)))))
42	39, 41	bitrid 192	. . . . 5 ⊢ (𝑛 = 𝐴 → (∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ ∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ 𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2)))))
43	42	2rexbidv 2575	. . . 4 ⊢ (𝑛 = 𝐴 → (∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ ∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ 𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2)))))
44	30, 43	bitrid 192	. . 3 ⊢ (𝑛 = 𝐴 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ ∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ 𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2)))))
45	19, 44	elab3 2978	. 2 ⊢ (𝐴 ∈ {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))} ↔ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ ∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ 𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))))
46	2, 45	bitri 184	1 ⊢ (𝐴 ∈ 𝑆 ↔ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ ∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ 𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  {cab 2224  ∃wrex 2529  Vcvv 2821  (class class class)co 6075   + caddc 8172  2c2 9334  ℕ₀cn0 9542  ℤcz 9623  ↑cexp 10953

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 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-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287

This theorem depends on definitions:  df-bi 117  df-dc 847  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-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901  df-seqfrec 10863  df-exp 10954

This theorem is referenced by:  4sqlem3  13147  4sqlem4  13149  4sqlem18  13165  4sq  13167