Theorem frec2uzlt2d 10819

Description: The mapping G (see frec2uz0d 10814) 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 10818	. 2 |- ( ph -> ( A e. B -> ( G ` A ) < ( G ` B ) ) )
6		nntri3or 6756	. . . 4 |- ( ( A e. om /\ B e. om ) -> ( A e. B \/ A = B \/ B e. A ) )
7	3, 4, 6	syl2anc 415	. . 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 5690	. . . . . . . . . 10 |- ( A = B -> ( G ` A ) = ( G ` B ) )
11	10	adantl 277	. . . . . . . . 9 |- ( ( ph /\ A = B ) -> ( G ` A ) = ( G ` B ) )
12	11	breq2d 4137	. . . . . . . 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 10815	. . . . . . . . . . 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 9747	. . . . . . . 8 |- ( ( ( ph /\ A = B ) /\ ( G ` A ) < ( G ` B ) ) -> ( G ` A ) e. RR )
18	17	ltnrd 8427	. . . . . . 7 |- ( ( ( ph /\ A = B ) /\ ( G ` A ) < ( G ` B ) ) -> -. ( G ` A ) < ( G ` A ) )
19	13, 18	pm2.21dd 629	. . . . . 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 10815	. . . . . . . . 9 |- ( ph -> ( G ` B ) e. ZZ )
23	22	adantr 276	. . . . . . . 8 |- ( ( ph /\ B e. A ) -> ( G ` B ) e. ZZ )
24	23	zred 9747	. . . . . . 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 9747	. . . . . . 7 |- ( ( ph /\ B e. A ) -> ( G ` A ) e. RR )
27	1, 2, 4, 3	frec2uzltd 10818	. . . . . . . 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 8436	. . . . . 6 |- ( ( ph /\ B e. A ) -> -. ( G ` A ) < ( G ` B ) )
30	29	pm2.21d 628	. . . . 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 1345	. . 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 1008   = wceq 1402   e. wcel 2209   class class class wbr 4125   |-> cmpt 4187   omcom 4732   ` cfv 5372  (class class class)co 6075  freccfrec 6651   1c1 8170   + caddc 8172   < clt 8350   ZZcz 9623

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-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  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-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-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-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-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-inn 9284  df-n0 9543  df-z 9624  df-uz 9901

This theorem is referenced by:  frec2uzisod  10822  frec2uzled  10844  nninfctlemfo  12795