Theorem phialllem1 4616

Description: Lemma for phiall 4618. Any set of numbers without zero is the Phi of a set. (Contributed by Scott Fenton, 14-Apr-2021.)

Hypothesis

Ref	Expression
phiall.1	|- A e. _V

Assertion

Ref	Expression
phialllem1	|- ((A C_ Nn /\ -. 0c e. A) -> E.x A = Phi x)

Distinct variable group:   x,A

Proof of Theorem phialllem1

Dummy variables y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2413	. . . . . . . . . . . . 13 |- (z = 0c -> (z e. A <-> 0c e. A))
2	1	biimpcd 215	. . . . . . . . . . . 12 |- (z e. A -> (z = 0c -> 0c e. A))
3	2	con3d 125	. . . . . . . . . . 11 |- (z e. A -> (-. 0c e. A -> -. z = 0c))
4	3	impcom 419	. . . . . . . . . 10 |- ((-. 0c e. A /\ z e. A) -> -. z = 0c)
5	4	adantll 694	. . . . . . . . 9 |- (((A C_ Nn /\ -. 0c e. A) /\ z e. A) -> -. z = 0c)
6		ssel2 3268	. . . . . . . . . . 11 |- ((A C_ Nn /\ z e. A) -> z e. Nn )
7	6	adantlr 695	. . . . . . . . . 10 |- (((A C_ Nn /\ -. 0c e. A) /\ z e. A) -> z e. Nn )
8		nnc0suc 4412	. . . . . . . . . 10 |- (z e. Nn <-> (z = 0c \/ E.x e. Nn z = (x +o 1c)))
9	7, 8	sylib 188	. . . . . . . . 9 |- (((A C_ Nn /\ -. 0c e. A) /\ z e. A) -> (z = 0c \/ E.x e. Nn z = (x +o 1c)))
10		orel1 371	. . . . . . . . 9 |- (-. z = 0c -> ((z = 0c \/ E.x e. Nn z = (x +o 1c)) -> E.x e. Nn z = (x +o 1c)))
11	5, 9, 10	sylc 56	. . . . . . . 8 |- (((A C_ Nn /\ -. 0c e. A) /\ z e. A) -> E.x e. Nn z = (x +o 1c))
12		anidm 625	. . . . . . . . . 10 |- ((z = (x +o 1c) /\ z = (x +o 1c)) <-> z = (x +o 1c))
13		eqeq1 2359	. . . . . . . . . . 11 |- (z = w -> (z = (x +o 1c) <-> w = (x +o 1c)))
14	13	anbi2d 684	. . . . . . . . . 10 |- (z = w -> ((z = (x +o 1c) /\ z = (x +o 1c)) <-> (z = (x +o 1c) /\ w = (x +o 1c))))
15	12, 14	syl5bbr 250	. . . . . . . . 9 |- (z = w -> (z = (x +o 1c) <-> (z = (x +o 1c) /\ w = (x +o 1c))))
16	15	rexbidv 2635	. . . . . . . 8 |- (z = w -> (E.x e. Nn z = (x +o 1c) <-> E.x e. Nn (z = (x +o 1c) /\ w = (x +o 1c))))
17	11, 16	syl5ibcom 211	. . . . . . 7 |- (((A C_ Nn /\ -. 0c e. A) /\ z e. A) -> (z = w -> E.x e. Nn (z = (x +o 1c) /\ w = (x +o 1c))))
18		eqtr3 2372	. . . . . . . 8 |- ((z = (x +o 1c) /\ w = (x +o 1c)) -> z = w)
19	18	rexlimivw 2734	. . . . . . 7 |- (E.x e. Nn (z = (x +o 1c) /\ w = (x +o 1c)) -> z = w)
20	17, 19	impbid1 194	. . . . . 6 |- (((A C_ Nn /\ -. 0c e. A) /\ z e. A) -> (z = w <-> E.x e. Nn (z = (x +o 1c) /\ w = (x +o 1c))))
21	20	rexbidva 2631	. . . . 5 |- ((A C_ Nn /\ -. 0c e. A) -> (E.z e. A z = w <-> E.z e. A E.x e. Nn (z = (x +o 1c) /\ w = (x +o 1c))))
22		risset 2661	. . . . 5 |- (w e. A <-> E.z e. A z = w)
23		rexcom 2772	. . . . 5 |- (E.x e. Nn E.z e. A (z = (x +o 1c) /\ w = (x +o 1c)) <-> E.z e. A E.x e. Nn (z = (x +o 1c) /\ w = (x +o 1c)))
24	21, 22, 23	3bitr4g 279	. . . 4 |- ((A C_ Nn /\ -. 0c e. A) -> (w e. A <-> E.x e. Nn E.z e. A (z = (x +o 1c) /\ w = (x +o 1c))))
25	24	abbi2dv 2468	. . 3 |- ((A C_ Nn /\ -. 0c e. A) -> A = {w | E.x e. Nn E.z e. A (z = (x +o 1c) /\ w = (x +o 1c))})
26		df-phi 4565	. . . 4 |- Phi {y e. Nn | E.z e. A z = (y +o 1c)} = {w | E.x e. {y e. Nn | E.z e. A z = (y +o 1c)}w = if(x e. Nn , (x +o 1c), x)}
27		addceq1 4383	. . . . . . . . 9 |- (y = x -> (y +o 1c) = (x +o 1c))
28	27	eqeq2d 2364	. . . . . . . 8 |- (y = x -> (z = (y +o 1c) <-> z = (x +o 1c)))
29	28	rexbidv 2635	. . . . . . 7 |- (y = x -> (E.z e. A z = (y +o 1c) <-> E.z e. A z = (x +o 1c)))
30	29	rexrab 3000	. . . . . 6 |- (E.x e. {y e. Nn | E.z e. A z = (y +o 1c)}w = if(x e. Nn , (x +o 1c), x) <-> E.x e. Nn (E.z e. A z = (x +o 1c) /\ w = if(x e. Nn , (x +o 1c), x)))
31		iftrue 3668	. . . . . . . . . 10 |- (x e. Nn -> if(x e. Nn , (x +o 1c), x) = (x +o 1c))
32	31	eqeq2d 2364	. . . . . . . . 9 |- (x e. Nn -> (w = if(x e. Nn , (x +o 1c), x) <-> w = (x +o 1c)))
33	32	anbi2d 684	. . . . . . . 8 |- (x e. Nn -> ((E.z e. A z = (x +o 1c) /\ w = if(x e. Nn , (x +o 1c), x)) <-> (E.z e. A z = (x +o 1c) /\ w = (x +o 1c))))
34		r19.41v 2764	. . . . . . . 8 |- (E.z e. A (z = (x +o 1c) /\ w = (x +o 1c)) <-> (E.z e. A z = (x +o 1c) /\ w = (x +o 1c)))
35	33, 34	syl6bbr 254	. . . . . . 7 |- (x e. Nn -> ((E.z e. A z = (x +o 1c) /\ w = if(x e. Nn , (x +o 1c), x)) <-> E.z e. A (z = (x +o 1c) /\ w = (x +o 1c))))
36	35	rexbiia 2647	. . . . . 6 |- (E.x e. Nn (E.z e. A z = (x +o 1c) /\ w = if(x e. Nn , (x +o 1c), x)) <-> E.x e. Nn E.z e. A (z = (x +o 1c) /\ w = (x +o 1c)))
37	30, 36	bitri 240	. . . . 5 |- (E.x e. {y e. Nn | E.z e. A z = (y +o 1c)}w = if(x e. Nn , (x +o 1c), x) <-> E.x e. Nn E.z e. A (z = (x +o 1c) /\ w = (x +o 1c)))
38	37	abbii 2465	. . . 4 |- {w | E.x e. {y e. Nn | E.z e. A z = (y +o 1c)}w = if(x e. Nn , (x +o 1c), x)} = {w | E.x e. Nn E.z e. A (z = (x +o 1c) /\ w = (x +o 1c))}
39	26, 38	eqtri 2373	. . 3 |- Phi {y e. Nn | E.z e. A z = (y +o 1c)} = {w | E.x e. Nn E.z e. A (z = (x +o 1c) /\ w = (x +o 1c))}
40	25, 39	syl6eqr 2403	. 2 |- ((A C_ Nn /\ -. 0c e. A) -> A = Phi {y e. Nn | E.z e. A z = (y +o 1c)})
41		dfrab2 3530	. . . 4 |- {y e. Nn | E.z e. A z = (y +o 1c)} = ({y | E.z e. A z = (y +o 1c)} i^i Nn )
42		vex 2862	. . . . . . . . 9 |- y e. _V
43	42	elimak 4259	. . . . . . . 8 |- (y e. (`'kImagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c)"kA) <-> E.z e. A <<z, y>> e. `'kImagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c))
44		vex 2862	. . . . . . . . . . 11 |- z e. _V
45	42, 44	opkelimagek 4272	. . . . . . . . . 10 |- (<<y, z>> e. Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c) <-> z = ((( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c)"ky))
46	44, 42	opkelcnvk 4250	. . . . . . . . . 10 |- (<<z, y>> e. `'kImagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c) <-> <<y, z>> e. Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c))
47		dfaddc2 4381	. . . . . . . . . . 11 |- (y +o 1c) = ((( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c)"ky)
48	47	eqeq2i 2363	. . . . . . . . . 10 |- (z = (y +o 1c) <-> z = ((( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c)"ky))
49	45, 46, 48	3bitr4i 268	. . . . . . . . 9 |- (<<z, y>> e. `'kImagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c) <-> z = (y +o 1c))
50	49	rexbii 2639	. . . . . . . 8 |- (E.z e. A <<z, y>> e. `'kImagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c) <-> E.z e. A z = (y +o 1c))
51	43, 50	bitri 240	. . . . . . 7 |- (y e. (`'kImagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c)"kA) <-> E.z e. A z = (y +o 1c))
52	51	abbi2i 2464	. . . . . 6 |- (`'kImagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c)"kA) = {y | E.z e. A z = (y +o 1c)}
53		addcexlem 4382	. . . . . . . . . 10 |- ( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c)) e. _V
54		1cex 4142	. . . . . . . . . . . 12 |- 1c e. _V
55	54	pw1ex 4303	. . . . . . . . . . 11 |- ~P1 1c e. _V
56	55	pw1ex 4303	. . . . . . . . . 10 |- ~P1 ~P1 1c e. _V
57	53, 56	imakex 4300	. . . . . . . . 9 |- (( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c) e. _V
58	57	imagekex 4312	. . . . . . . 8 |- Imagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c) e. _V
59	58	cnvkex 4287	. . . . . . 7 |- `'kImagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c) e. _V
60		phiall.1	. . . . . . 7 |- A e. _V
61	59, 60	imakex 4300	. . . . . 6 |- (`'kImagek(( Ins3k ∼ (( Ins3k Sk i^i Ins2k Sk )"k~P1 ~P1 1c) \ (( Ins2k Ins2k Sk (+) ( Ins2k Ins3k Sk u. Ins3k SIk SIk Sk ))"k~P1 ~P1 ~P1 ~P1 1c))"k~P1 ~P1 1c)"kA) e. _V
62	52, 61	eqeltrri 2424	. . . . 5 |- {y | E.z e. A z = (y +o 1c)} e. _V
63		nncex 4396	. . . . 5 |- Nn e. _V
64	62, 63	inex 4105	. . . 4 |- ({y | E.z e. A z = (y +o 1c)} i^i Nn ) e. _V
65	41, 64	eqeltri 2423	. . 3 |- {y e. Nn | E.z e. A z = (y +o 1c)} e. _V
66		phieq 4570	. . . 4 |- (x = {y e. Nn | E.z e. A z = (y +o 1c)} -> Phi x = Phi {y e. Nn | E.z e. A z = (y +o 1c)})
67	66	eqeq2d 2364	. . 3 |- (x = {y e. Nn | E.z e. A z = (y +o 1c)} -> (A = Phi x <-> A = Phi {y e. Nn | E.z e. A z = (y +o 1c)}))
68	65, 67	spcev 2946	. 2 |- (A = Phi {y e. Nn | E.z e. A z = (y +o 1c)} -> E.x A = Phi x)
69	40, 68	syl 15	1 |- ((A C_ Nn /\ -. 0c e. A) -> E.x A = Phi x)

Syntax hints:  -. wn 3   -> wi 4   \/ wo 357   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339  E.wrex 2615  {crab 2618  _Vcvv 2859   ∼ ccompl 3205   \ cdif 3206   u. cun 3207   i^i cin 3208   (+) csymdif 3209   C_ wss 3257  ifcif 3662  <<copk 4057  1_cc1c 4134  ~P₁ cpw1 4135  `'_kccnvk 4175   Ins2_k cins2k 4176   Ins3_k cins3k 4177  "_kcimak 4179   SI_k csik 4181  Image_kcimagek 4182   S_k cssetk 4183   Nn cnnc 4373  0_cc0c 4374   +o cplc 4375   Phi cphi 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-0c 4377  df-addc 4378  df-nnc 4379  df-phi 4565

This theorem is referenced by:  phialllem2  4617