Theorem cantnfp1 8578

Description: If F is created by adding a single term ( F ` X ) = Y to G, where X is larger than any element of the support of G, then F is also a finitely supported function and it is assigned the value ( ( A ^o X ) .o Y ) +o z where z is the value of G. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 1-Jul-2019.)

Hypotheses

Ref	Expression
cantnfs.s	|- S = dom ( A CNF B )
cantnfs.a	|- ( ph -> A e. On )
cantnfs.b	|- ( ph -> B e. On )
cantnfp1.g	|- ( ph -> G e. S )
cantnfp1.x	|- ( ph -> X e. B )
cantnfp1.y	|- ( ph -> Y e. A )
cantnfp1.s	|- ( ph -> ( G supp (/) ) C_ X )
cantnfp1.f	|- F = ( t e. B |-> if ( t = X , Y , ( G ` t ) ) )

Assertion

Ref	Expression
cantnfp1	|- ( ph -> ( F e. S /\ ( ( A CNF B ) ` F ) = ( ( ( A ^o X ) .o Y ) +o ( ( A CNF B ) ` G ) ) ) )

Distinct variable groups:   t, B   t, A   t, S   t, G   ph, t   t, Y   t, X

Allowed substitution hint:   F( t)

Proof of Theorem cantnfp1

Dummy variables k z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cantnfp1.f	. . . . . 6 |- F = ( t e. B |-> if ( t = X , Y , ( G ` t ) ) )
2		cantnfs.b	. . . . . . . . . . . . 13 |- ( ph -> B e. On )
3		cantnfp1.x	. . . . . . . . . . . . 13 |- ( ph -> X e. B )
4		onelon 5748	. . . . . . . . . . . . 13 |- ( ( B e. On /\ X e. B ) -> X e. On )
5	2, 3, 4	syl2anc 693	. . . . . . . . . . . 12 |- ( ph -> X e. On )
6		eloni 5733	. . . . . . . . . . . 12 |- ( X e. On -> Ord X )
7		ordirr 5741	. . . . . . . . . . . 12 |- ( Ord X -> -. X e. X )
8	5, 6, 7	3syl 18	. . . . . . . . . . 11 |- ( ph -> -. X e. X )
9		fvex 6201	. . . . . . . . . . . . . 14 |- ( G ` X ) e. _V
10		dif1o 7580	. . . . . . . . . . . . . 14 |- ( ( G ` X ) e. ( _V \ 1o ) <-> ( ( G ` X ) e. _V /\ ( G ` X ) =/= (/) ) )
11	9, 10	mpbiran 953	. . . . . . . . . . . . 13 |- ( ( G ` X ) e. ( _V \ 1o ) <-> ( G ` X ) =/= (/) )
12		cantnfp1.g	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> G e. S )
13		cantnfs.s	. . . . . . . . . . . . . . . . . . . . 21 |- S = dom ( A CNF B )
14		cantnfs.a	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> A e. On )
15	13, 14, 2	cantnfs 8563	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( G e. S <-> ( G : B --> A /\ G finSupp (/) ) ) )
16	12, 15	mpbid 222	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( G : B --> A /\ G finSupp (/) ) )
17	16	simpld 475	. . . . . . . . . . . . . . . . . 18 |- ( ph -> G : B --> A )
18		ffn 6045	. . . . . . . . . . . . . . . . . 18 |- ( G : B --> A -> G Fn B )
19	17, 18	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ph -> G Fn B )
20		0ex 4790	. . . . . . . . . . . . . . . . . 18 |- (/) e. _V
21	20	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ph -> (/) e. _V )
22		elsuppfn 7303	. . . . . . . . . . . . . . . . 17 |- ( ( G Fn B /\ B e. On /\ (/) e. _V ) -> ( X e. ( G supp (/) ) <-> ( X e. B /\ ( G ` X ) =/= (/) ) ) )
23	19, 2, 21, 22	syl3anc 1326	. . . . . . . . . . . . . . . 16 |- ( ph -> ( X e. ( G supp (/) ) <-> ( X e. B /\ ( G ` X ) =/= (/) ) ) )
24	11	bicomi 214	. . . . . . . . . . . . . . . . . 18 |- ( ( G ` X ) =/= (/) <-> ( G ` X ) e. ( _V \ 1o ) )
25	24	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( ( G ` X ) =/= (/) <-> ( G ` X ) e. ( _V \ 1o ) ) )
26	25	anbi2d 740	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ( X e. B /\ ( G ` X ) =/= (/) ) <-> ( X e. B /\ ( G ` X ) e. ( _V \ 1o ) ) ) )
27	23, 26	bitrd 268	. . . . . . . . . . . . . . 15 |- ( ph -> ( X e. ( G supp (/) ) <-> ( X e. B /\ ( G ` X ) e. ( _V \ 1o ) ) ) )
28		cantnfp1.s	. . . . . . . . . . . . . . . 16 |- ( ph -> ( G supp (/) ) C_ X )
29	28	sseld 3602	. . . . . . . . . . . . . . 15 |- ( ph -> ( X e. ( G supp (/) ) -> X e. X ) )
30	27, 29	sylbird 250	. . . . . . . . . . . . . 14 |- ( ph -> ( ( X e. B /\ ( G ` X ) e. ( _V \ 1o ) ) -> X e. X ) )
31	3, 30	mpand 711	. . . . . . . . . . . . 13 |- ( ph -> ( ( G ` X ) e. ( _V \ 1o ) -> X e. X ) )
32	11, 31	syl5bir 233	. . . . . . . . . . . 12 |- ( ph -> ( ( G ` X ) =/= (/) -> X e. X ) )
33	32	necon1bd 2812	. . . . . . . . . . 11 |- ( ph -> ( -. X e. X -> ( G ` X ) = (/) ) )
34	8, 33	mpd 15	. . . . . . . . . 10 |- ( ph -> ( G ` X ) = (/) )
35	34	ad3antrrr 766	. . . . . . . . 9 |- ( ( ( ( ph /\ Y = (/) ) /\ t e. B ) /\ t = X ) -> ( G ` X ) = (/) )
36		simpr 477	. . . . . . . . . 10 |- ( ( ( ( ph /\ Y = (/) ) /\ t e. B ) /\ t = X ) -> t = X )
37	36	fveq2d 6195	. . . . . . . . 9 |- ( ( ( ( ph /\ Y = (/) ) /\ t e. B ) /\ t = X ) -> ( G ` t ) = ( G ` X ) )
38		simpllr 799	. . . . . . . . 9 |- ( ( ( ( ph /\ Y = (/) ) /\ t e. B ) /\ t = X ) -> Y = (/) )
39	35, 37, 38	3eqtr4rd 2667	. . . . . . . 8 |- ( ( ( ( ph /\ Y = (/) ) /\ t e. B ) /\ t = X ) -> Y = ( G ` t ) )
40		eqidd 2623	. . . . . . . 8 |- ( ( ( ( ph /\ Y = (/) ) /\ t e. B ) /\ -. t = X ) -> ( G ` t ) = ( G ` t ) )
41	39, 40	ifeqda 4121	. . . . . . 7 |- ( ( ( ph /\ Y = (/) ) /\ t e. B ) -> if ( t = X , Y , ( G ` t ) ) = ( G ` t ) )
42	41	mpteq2dva 4744	. . . . . 6 |- ( ( ph /\ Y = (/) ) -> ( t e. B |-> if ( t = X , Y , ( G ` t ) ) ) = ( t e. B |-> ( G ` t ) ) )
43	1, 42	syl5eq 2668	. . . . 5 |- ( ( ph /\ Y = (/) ) -> F = ( t e. B |-> ( G ` t ) ) )
44	17	feqmptd 6249	. . . . . 6 |- ( ph -> G = ( t e. B |-> ( G ` t ) ) )
45	44	adantr 481	. . . . 5 |- ( ( ph /\ Y = (/) ) -> G = ( t e. B |-> ( G ` t ) ) )
46	43, 45	eqtr4d 2659	. . . 4 |- ( ( ph /\ Y = (/) ) -> F = G )
47	12	adantr 481	. . . 4 |- ( ( ph /\ Y = (/) ) -> G e. S )
48	46, 47	eqeltrd 2701	. . 3 |- ( ( ph /\ Y = (/) ) -> F e. S )
49		oecl 7617	. . . . . . . 8 |- ( ( A e. On /\ B e. On ) -> ( A ^o B ) e. On )
50	14, 2, 49	syl2anc 693	. . . . . . 7 |- ( ph -> ( A ^o B ) e. On )
51	13, 14, 2	cantnff 8571	. . . . . . . 8 |- ( ph -> ( A CNF B ) : S --> ( A ^o B ) )
52	51, 12	ffvelrnd 6360	. . . . . . 7 |- ( ph -> ( ( A CNF B ) ` G ) e. ( A ^o B ) )
53		onelon 5748	. . . . . . 7 |- ( ( ( A ^o B ) e. On /\ ( ( A CNF B ) ` G ) e. ( A ^o B ) ) -> ( ( A CNF B ) ` G ) e. On )
54	50, 52, 53	syl2anc 693	. . . . . 6 |- ( ph -> ( ( A CNF B ) ` G ) e. On )
55	54	adantr 481	. . . . 5 |- ( ( ph /\ Y = (/) ) -> ( ( A CNF B ) ` G ) e. On )
56		oa0r 7618	. . . . 5 |- ( ( ( A CNF B ) ` G ) e. On -> ( (/) +o ( ( A CNF B ) ` G ) ) = ( ( A CNF B ) ` G ) )
57	55, 56	syl 17	. . . 4 |- ( ( ph /\ Y = (/) ) -> ( (/) +o ( ( A CNF B ) ` G ) ) = ( ( A CNF B ) ` G ) )
58		oveq2 6658	. . . . . 6 |- ( Y = (/) -> ( ( A ^o X ) .o Y ) = ( ( A ^o X ) .o (/) ) )
59		oecl 7617	. . . . . . . 8 |- ( ( A e. On /\ X e. On ) -> ( A ^o X ) e. On )
60	14, 5, 59	syl2anc 693	. . . . . . 7 |- ( ph -> ( A ^o X ) e. On )
61		om0 7597	. . . . . . 7 |- ( ( A ^o X ) e. On -> ( ( A ^o X ) .o (/) ) = (/) )
62	60, 61	syl 17	. . . . . 6 |- ( ph -> ( ( A ^o X ) .o (/) ) = (/) )
63	58, 62	sylan9eqr 2678	. . . . 5 |- ( ( ph /\ Y = (/) ) -> ( ( A ^o X ) .o Y ) = (/) )
64	63	oveq1d 6665	. . . 4 |- ( ( ph /\ Y = (/) ) -> ( ( ( A ^o X ) .o Y ) +o ( ( A CNF B ) ` G ) ) = ( (/) +o ( ( A CNF B ) ` G ) ) )
65	46	fveq2d 6195	. . . 4 |- ( ( ph /\ Y = (/) ) -> ( ( A CNF B ) ` F ) = ( ( A CNF B ) ` G ) )
66	57, 64, 65	3eqtr4rd 2667	. . 3 |- ( ( ph /\ Y = (/) ) -> ( ( A CNF B ) ` F ) = ( ( ( A ^o X ) .o Y ) +o ( ( A CNF B ) ` G ) ) )
67	48, 66	jca 554	. 2 |- ( ( ph /\ Y = (/) ) -> ( F e. S /\ ( ( A CNF B ) ` F ) = ( ( ( A ^o X ) .o Y ) +o ( ( A CNF B ) ` G ) ) ) )
68	14	adantr 481	. . . 4 |- ( ( ph /\ Y =/= (/) ) -> A e. On )
69	2	adantr 481	. . . 4 |- ( ( ph /\ Y =/= (/) ) -> B e. On )
70	12	adantr 481	. . . 4 |- ( ( ph /\ Y =/= (/) ) -> G e. S )
71	3	adantr 481	. . . 4 |- ( ( ph /\ Y =/= (/) ) -> X e. B )
72		cantnfp1.y	. . . . 5 |- ( ph -> Y e. A )
73	72	adantr 481	. . . 4 |- ( ( ph /\ Y =/= (/) ) -> Y e. A )
74	28	adantr 481	. . . 4 |- ( ( ph /\ Y =/= (/) ) -> ( G supp (/) ) C_ X )
75	13, 68, 69, 70, 71, 73, 74, 1	cantnfp1lem1 8575	. . 3 |- ( ( ph /\ Y =/= (/) ) -> F e. S )
76		onelon 5748	. . . . . . 7 |- ( ( A e. On /\ Y e. A ) -> Y e. On )
77	14, 72, 76	syl2anc 693	. . . . . 6 |- ( ph -> Y e. On )
78		on0eln0 5780	. . . . . 6 |- ( Y e. On -> ( (/) e. Y <-> Y =/= (/) ) )
79	77, 78	syl 17	. . . . 5 |- ( ph -> ( (/) e. Y <-> Y =/= (/) ) )
80	79	biimpar 502	. . . 4 |- ( ( ph /\ Y =/= (/) ) -> (/) e. Y )
81		eqid 2622	. . . 4 |- OrdIso ( _E , ( F supp (/) ) ) = OrdIso ( _E , ( F supp (/) ) )
82		eqid 2622	. . . 4 |- seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o (OrdIso ( _E , ( F supp (/) ) ) ` k ) ) .o ( F ` (OrdIso ( _E , ( F supp (/) ) ) ` k ) ) ) +o z ) ) , (/) ) = seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o (OrdIso ( _E , ( F supp (/) ) ) ` k ) ) .o ( F ` (OrdIso ( _E , ( F supp (/) ) ) ` k ) ) ) +o z ) ) , (/) )
83		eqid 2622	. . . 4 |- OrdIso ( _E , ( G supp (/) ) ) = OrdIso ( _E , ( G supp (/) ) )
84		eqid 2622	. . . 4 |- seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o (OrdIso ( _E , ( G supp (/) ) ) ` k ) ) .o ( G ` (OrdIso ( _E , ( G supp (/) ) ) ` k ) ) ) +o z ) ) , (/) ) = seq𝜔 ( ( k e. _V , z e. _V |-> ( ( ( A ^o (OrdIso ( _E , ( G supp (/) ) ) ` k ) ) .o ( G ` (OrdIso ( _E , ( G supp (/) ) ) ` k ) ) ) +o z ) ) , (/) )
85	13, 68, 69, 70, 71, 73, 74, 1, 80, 81, 82, 83, 84	cantnfp1lem3 8577	. . 3 |- ( ( ph /\ Y =/= (/) ) -> ( ( A CNF B ) ` F ) = ( ( ( A ^o X ) .o Y ) +o ( ( A CNF B ) ` G ) ) )
86	75, 85	jca 554	. 2 |- ( ( ph /\ Y =/= (/) ) -> ( F e. S /\ ( ( A CNF B ) ` F ) = ( ( ( A ^o X ) .o Y ) +o ( ( A CNF B ) ` G ) ) ) )
87	67, 86	pm2.61dane 2881	1 |- ( ph -> ( F e. S /\ ( ( A CNF B ) ` F ) = ( ( ( A ^o X ) .o Y ) +o ( ( A CNF B ) ` G ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   _E cep 5028   dom cdm 5114   Ord word 5722   Oncon0 5723   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   supp csupp 7295  seq_𝜔cseqom 7542   1oc1o 7553   +o coa 7557   .o comu 7558   ^o coe 7559   finSupp cfsupp 8275  OrdIsocoi 8414   CNF ccnf 8558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-cnf 8559

This theorem is referenced by:  cantnflem1d  8585  cantnflem1  8586  cantnflem3  8588