Theorem nolesgn2o 31824

Description: Given A less than or equal to B, equal to B up to X, and A ( X ) = 2o, then B ( X ) = 2o. (Contributed by Scott Fenton, 6-Dec-2021.)

Assertion

Ref	Expression
nolesgn2o	|- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> ( B ` X ) = 2o )

Proof of Theorem nolesgn2o

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl2 1065	. . . . . 6 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) ) -> B e. No )
2		nofv 31810	. . . . . 6 |- ( B e. No -> ( ( B ` X ) = (/) \/ ( B ` X ) = 1o \/ ( B ` X ) = 2o ) )
3	1, 2	syl 17	. . . . 5 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) ) -> ( ( B ` X ) = (/) \/ ( B ` X ) = 1o \/ ( B ` X ) = 2o ) )
4		3orel3 31593	. . . . 5 |- ( -. ( B ` X ) = 2o -> ( ( ( B ` X ) = (/) \/ ( B ` X ) = 1o \/ ( B ` X ) = 2o ) -> ( ( B ` X ) = (/) \/ ( B ` X ) = 1o ) ) )
5	3, 4	syl5com 31	. . . 4 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) ) -> ( -. ( B ` X ) = 2o -> ( ( B ` X ) = (/) \/ ( B ` X ) = 1o ) ) )
6		simp13 1093	. . . . . . 7 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ ( ( B ` X ) = (/) \/ ( B ` X ) = 1o ) ) -> X e. On )
7		fveq1 6190	. . . . . . . . . . . . 13 |- ( ( A |` X ) = ( B |` X ) -> ( ( A |` X ) ` y ) = ( ( B |` X ) ` y ) )
8	7	eqcomd 2628	. . . . . . . . . . . 12 |- ( ( A |` X ) = ( B |` X ) -> ( ( B |` X ) ` y ) = ( ( A |` X ) ` y ) )
9	8	adantr 481	. . . . . . . . . . 11 |- ( ( ( A |` X ) = ( B |` X ) /\ y e. X ) -> ( ( B |` X ) ` y ) = ( ( A |` X ) ` y ) )
10		simpr 477	. . . . . . . . . . . 12 |- ( ( ( A |` X ) = ( B |` X ) /\ y e. X ) -> y e. X )
11	10	fvresd 6208	. . . . . . . . . . 11 |- ( ( ( A |` X ) = ( B |` X ) /\ y e. X ) -> ( ( B |` X ) ` y ) = ( B ` y ) )
12	10	fvresd 6208	. . . . . . . . . . 11 |- ( ( ( A |` X ) = ( B |` X ) /\ y e. X ) -> ( ( A |` X ) ` y ) = ( A ` y ) )
13	9, 11, 12	3eqtr3d 2664	. . . . . . . . . 10 |- ( ( ( A |` X ) = ( B |` X ) /\ y e. X ) -> ( B ` y ) = ( A ` y ) )
14	13	ralrimiva 2966	. . . . . . . . 9 |- ( ( A |` X ) = ( B |` X ) -> A. y e. X ( B ` y ) = ( A ` y ) )
15	14	adantr 481	. . . . . . . 8 |- ( ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) -> A. y e. X ( B ` y ) = ( A ` y ) )
16	15	3ad2ant2 1083	. . . . . . 7 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ ( ( B ` X ) = (/) \/ ( B ` X ) = 1o ) ) -> A. y e. X ( B ` y ) = ( A ` y ) )
17		simprr 796	. . . . . . . . . . . . 13 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) ) -> ( A ` X ) = 2o )
18	17	a1d 25	. . . . . . . . . . . 12 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) ) -> ( ( B ` X ) = (/) -> ( A ` X ) = 2o ) )
19	18	ancld 576	. . . . . . . . . . 11 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) ) -> ( ( B ` X ) = (/) -> ( ( B ` X ) = (/) /\ ( A ` X ) = 2o ) ) )
20	17	a1d 25	. . . . . . . . . . . 12 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) ) -> ( ( B ` X ) = 1o -> ( A ` X ) = 2o ) )
21	20	ancld 576	. . . . . . . . . . 11 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) ) -> ( ( B ` X ) = 1o -> ( ( B ` X ) = 1o /\ ( A ` X ) = 2o ) ) )
22	19, 21	orim12d 883	. . . . . . . . . 10 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) ) -> ( ( ( B ` X ) = (/) \/ ( B ` X ) = 1o ) -> ( ( ( B ` X ) = (/) /\ ( A ` X ) = 2o ) \/ ( ( B ` X ) = 1o /\ ( A ` X ) = 2o ) ) ) )
23	22	3impia 1261	. . . . . . . . 9 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ ( ( B ` X ) = (/) \/ ( B ` X ) = 1o ) ) -> ( ( ( B ` X ) = (/) /\ ( A ` X ) = 2o ) \/ ( ( B ` X ) = 1o /\ ( A ` X ) = 2o ) ) )
24		3mix3 1232	. . . . . . . . . 10 |- ( ( ( B ` X ) = (/) /\ ( A ` X ) = 2o ) -> ( ( ( B ` X ) = 1o /\ ( A ` X ) = (/) ) \/ ( ( B ` X ) = 1o /\ ( A ` X ) = 2o ) \/ ( ( B ` X ) = (/) /\ ( A ` X ) = 2o ) ) )
25		3mix2 1231	. . . . . . . . . 10 |- ( ( ( B ` X ) = 1o /\ ( A ` X ) = 2o ) -> ( ( ( B ` X ) = 1o /\ ( A ` X ) = (/) ) \/ ( ( B ` X ) = 1o /\ ( A ` X ) = 2o ) \/ ( ( B ` X ) = (/) /\ ( A ` X ) = 2o ) ) )
26	24, 25	jaoi 394	. . . . . . . . 9 |- ( ( ( ( B ` X ) = (/) /\ ( A ` X ) = 2o ) \/ ( ( B ` X ) = 1o /\ ( A ` X ) = 2o ) ) -> ( ( ( B ` X ) = 1o /\ ( A ` X ) = (/) ) \/ ( ( B ` X ) = 1o /\ ( A ` X ) = 2o ) \/ ( ( B ` X ) = (/) /\ ( A ` X ) = 2o ) ) )
27	23, 26	syl 17	. . . . . . . 8 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ ( ( B ` X ) = (/) \/ ( B ` X ) = 1o ) ) -> ( ( ( B ` X ) = 1o /\ ( A ` X ) = (/) ) \/ ( ( B ` X ) = 1o /\ ( A ` X ) = 2o ) \/ ( ( B ` X ) = (/) /\ ( A ` X ) = 2o ) ) )
28		fvex 6201	. . . . . . . . 9 |- ( B ` X ) e. _V
29		fvex 6201	. . . . . . . . 9 |- ( A ` X ) e. _V
30	28, 29	brtp 31639	. . . . . . . 8 |- ( ( B ` X ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( A ` X ) <-> ( ( ( B ` X ) = 1o /\ ( A ` X ) = (/) ) \/ ( ( B ` X ) = 1o /\ ( A ` X ) = 2o ) \/ ( ( B ` X ) = (/) /\ ( A ` X ) = 2o ) ) )
31	27, 30	sylibr 224	. . . . . . 7 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ ( ( B ` X ) = (/) \/ ( B ` X ) = 1o ) ) -> ( B ` X ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( A ` X ) )
32		raleq 3138	. . . . . . . . 9 |- ( x = X -> ( A. y e. x ( B ` y ) = ( A ` y ) <-> A. y e. X ( B ` y ) = ( A ` y ) ) )
33		fveq2 6191	. . . . . . . . . 10 |- ( x = X -> ( B ` x ) = ( B ` X ) )
34		fveq2 6191	. . . . . . . . . 10 |- ( x = X -> ( A ` x ) = ( A ` X ) )
35	33, 34	breq12d 4666	. . . . . . . . 9 |- ( x = X -> ( ( B ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( A ` x ) <-> ( B ` X ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( A ` X ) ) )
36	32, 35	anbi12d 747	. . . . . . . 8 |- ( x = X -> ( ( A. y e. x ( B ` y ) = ( A ` y ) /\ ( B ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( A ` x ) ) <-> ( A. y e. X ( B ` y ) = ( A ` y ) /\ ( B ` X ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( A ` X ) ) ) )
37	36	rspcev 3309	. . . . . . 7 |- ( ( X e. On /\ ( A. y e. X ( B ` y ) = ( A ` y ) /\ ( B ` X ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( A ` X ) ) ) -> E. x e. On ( A. y e. x ( B ` y ) = ( A ` y ) /\ ( B ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( A ` x ) ) )
38	6, 16, 31, 37	syl12anc 1324	. . . . . 6 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ ( ( B ` X ) = (/) \/ ( B ` X ) = 1o ) ) -> E. x e. On ( A. y e. x ( B ` y ) = ( A ` y ) /\ ( B ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( A ` x ) ) )
39		simp12 1092	. . . . . . 7 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ ( ( B ` X ) = (/) \/ ( B ` X ) = 1o ) ) -> B e. No )
40		simp11 1091	. . . . . . 7 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ ( ( B ` X ) = (/) \/ ( B ` X ) = 1o ) ) -> A e. No )
41		sltval 31800	. . . . . . 7 |- ( ( B e. No /\ A e. No ) -> ( B <s A <-> E. x e. On ( A. y e. x ( B ` y ) = ( A ` y ) /\ ( B ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( A ` x ) ) ) )
42	39, 40, 41	syl2anc 693	. . . . . 6 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ ( ( B ` X ) = (/) \/ ( B ` X ) = 1o ) ) -> ( B <s A <-> E. x e. On ( A. y e. x ( B ` y ) = ( A ` y ) /\ ( B ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( A ` x ) ) ) )
43	38, 42	mpbird 247	. . . . 5 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ ( ( B ` X ) = (/) \/ ( B ` X ) = 1o ) ) -> B <s A )
44	43	3expia 1267	. . . 4 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) ) -> ( ( ( B ` X ) = (/) \/ ( B ` X ) = 1o ) -> B <s A ) )
45	5, 44	syld 47	. . 3 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) ) -> ( -. ( B ` X ) = 2o -> B <s A ) )
46	45	con1d 139	. 2 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) ) -> ( -. B <s A -> ( B ` X ) = 2o ) )
47	46	3impia 1261	1 |- ( ( ( A e. No /\ B e. No /\ X e. On ) /\ ( ( A |` X ) = ( B |` X ) /\ ( A ` X ) = 2o ) /\ -. B <s A ) -> ( B ` X ) = 2o )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   (/)c0 3915   {ctp 4181   <.cop 4183   class class class wbr 4653   |` cres 5116   Oncon0 5723   ` cfv 5888   1oc1o 7553   2oc2o 7554   Nocsur 31793   <scslt 31794

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-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-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-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-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-ord 5726  df-on 5727  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-1o 7560  df-2o 7561  df-no 31796  df-slt 31797

This theorem is referenced by:  nolesgn2ores  31825