Theorem frec2uzlt2d 10621

Description: The mapping G (see frec2uz0d 10616) preserves order. (Contributed by Jim Kingdon, 16-May-2020.)

Hypotheses

Ref	Expression
frec2uz.1	|- ( ph -> C e. ZZ )
frec2uz.2	|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )
frec2uzzd.a	|- ( ph -> A e. om )
frec2uzltd.b	|- ( ph -> B e. om )

Assertion

Ref	Expression
frec2uzlt2d	|- ( ph -> ( A e. B <-> ( G ` A ) < ( G ` B ) ) )

Distinct variable group:   x, C

Allowed substitution hints:   ph( x)   A( x)   B( x)   G( x)

Proof of Theorem frec2uzlt2d

Step	Hyp	Ref	Expression
1		frec2uz.1	. . 3 |- ( ph -> C e. ZZ )
2		frec2uz.2	. . 3 |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )
3		frec2uzzd.a	. . 3 |- ( ph -> A e. om )
4		frec2uzltd.b	. . 3 |- ( ph -> B e. om )
5	1, 2, 3, 4	frec2uzltd 10620	. 2 |- ( ph -> ( A e. B -> ( G ` A ) < ( G ` B ) ) )
6		nntri3or 6637	. . . 4 |- ( ( A e. om /\ B e. om ) -> ( A e. B \/ A = B \/ B e. A ) )
7	3, 4, 6	syl2anc 411	. . 3 |- ( ph -> ( A e. B \/ A = B \/ B e. A ) )
8		ax-1 6	. . . . 5 |- ( A e. B -> ( ( G ` A ) < ( G ` B ) -> A e. B ) )
9	8	a1i 9	. . . 4 |- ( ph -> ( A e. B -> ( ( G ` A ) < ( G ` B ) -> A e. B ) ) )
10		fveq2 5626	. . . . . . . . . 10 |- ( A = B -> ( G ` A ) = ( G ` B ) )
11	10	adantl 277	. . . . . . . . 9 |- ( ( ph /\ A = B ) -> ( G ` A ) = ( G ` B ) )
12	11	breq2d 4094	. . . . . . . 8 |- ( ( ph /\ A = B ) -> ( ( G ` A ) < ( G ` A ) <-> ( G ` A ) < ( G ` B ) ) )
13	12	biimpar 297	. . . . . . 7 |- ( ( ( ph /\ A = B ) /\ ( G ` A ) < ( G ` B ) ) -> ( G ` A ) < ( G ` A ) )
14	1, 2, 3	frec2uzzd 10617	. . . . . . . . . . 11 |- ( ph -> ( G ` A ) e. ZZ )
15	14	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ A = B ) -> ( G ` A ) e. ZZ )
16	15	adantr 276	. . . . . . . . 9 |- ( ( ( ph /\ A = B ) /\ ( G ` A ) < ( G ` B ) ) -> ( G ` A ) e. ZZ )
17	16	zred 9565	. . . . . . . 8 |- ( ( ( ph /\ A = B ) /\ ( G ` A ) < ( G ` B ) ) -> ( G ` A ) e. RR )
18	17	ltnrd 8254	. . . . . . 7 |- ( ( ( ph /\ A = B ) /\ ( G ` A ) < ( G ` B ) ) -> -. ( G ` A ) < ( G ` A ) )
19	13, 18	pm2.21dd 623	. . . . . 6 |- ( ( ( ph /\ A = B ) /\ ( G ` A ) < ( G ` B ) ) -> A e. B )
20	19	ex 115	. . . . 5 |- ( ( ph /\ A = B ) -> ( ( G ` A ) < ( G ` B ) -> A e. B ) )
21	20	ex 115	. . . 4 |- ( ph -> ( A = B -> ( ( G ` A ) < ( G ` B ) -> A e. B ) ) )
22	1, 2, 4	frec2uzzd 10617	. . . . . . . . 9 |- ( ph -> ( G ` B ) e. ZZ )
23	22	adantr 276	. . . . . . . 8 |- ( ( ph /\ B e. A ) -> ( G ` B ) e. ZZ )
24	23	zred 9565	. . . . . . 7 |- ( ( ph /\ B e. A ) -> ( G ` B ) e. RR )
25	14	adantr 276	. . . . . . . 8 |- ( ( ph /\ B e. A ) -> ( G ` A ) e. ZZ )
26	25	zred 9565	. . . . . . 7 |- ( ( ph /\ B e. A ) -> ( G ` A ) e. RR )
27	1, 2, 4, 3	frec2uzltd 10620	. . . . . . . 8 |- ( ph -> ( B e. A -> ( G ` B ) < ( G ` A ) ) )
28	27	imp 124	. . . . . . 7 |- ( ( ph /\ B e. A ) -> ( G ` B ) < ( G ` A ) )
29	24, 26, 28	ltnsymd 8262	. . . . . 6 |- ( ( ph /\ B e. A ) -> -. ( G ` A ) < ( G ` B ) )
30	29	pm2.21d 622	. . . . 5 |- ( ( ph /\ B e. A ) -> ( ( G ` A ) < ( G ` B ) -> A e. B ) )
31	30	ex 115	. . . 4 |- ( ph -> ( B e. A -> ( ( G ` A ) < ( G ` B ) -> A e. B ) ) )
32	9, 21, 31	3jaod 1338	. . 3 |- ( ph -> ( ( A e. B \/ A = B \/ B e. A ) -> ( ( G ` A ) < ( G ` B ) -> A e. B ) ) )
33	7, 32	mpd 13	. 2 |- ( ph -> ( ( G ` A ) < ( G ` B ) -> A e. B ) )
34	5, 33	impbid 129	1 |- ( ph -> ( A e. B <-> ( G ` A ) < ( G ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 1001   = wceq 1395   e. wcel 2200   class class class wbr 4082   |-> cmpt 4144   omcom 4681   ` cfv 5317  (class class class)co 6000  freccfrec 6534   1c1 7996   + caddc 7998   < clt 8177   ZZcz 9442

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-recs 6449  df-frec 6535  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-n0 9366  df-z 9443  df-uz 9719

This theorem is referenced by:  frec2uzisod  10624  frec2uzled  10646  nninfctlemfo  12556