Theorem recvguniqlem 11176

Description: Lemma for recvguniq 11177. Some of the rearrangements of the expressions. (Contributed by Jim Kingdon, 8-Aug-2021.)

Hypotheses

Ref	Expression
recvguniqlem.f	|- ( ph -> F : NN --> RR )
recvguniqlem.a	|- ( ph -> A e. RR )
recvguniqlem.b	|- ( ph -> B e. RR )
recvguniqlem.k	|- ( ph -> K e. NN )
recvguniqlem.lt1	|- ( ph -> A < ( ( F ` K ) + ( ( A - B ) / 2 ) ) )
recvguniqlem.lt2	|- ( ph -> ( F ` K ) < ( B + ( ( A - B ) / 2 ) ) )

Assertion

Ref	Expression
recvguniqlem	|- ( ph -> F. )

Proof of Theorem recvguniqlem

Step	Hyp	Ref	Expression
1		recvguniqlem.a	. . 3 |- ( ph -> A e. RR )
2		recvguniqlem.f	. . . . 5 |- ( ph -> F : NN --> RR )
3		recvguniqlem.k	. . . . 5 |- ( ph -> K e. NN )
4	2, 3	ffvelcdmd 5701	. . . 4 |- ( ph -> ( F ` K ) e. RR )
5		recvguniqlem.b	. . . . . 6 |- ( ph -> B e. RR )
6	1, 5	resubcld 8424	. . . . 5 |- ( ph -> ( A - B ) e. RR )
7	6	rehalfcld 9255	. . . 4 |- ( ph -> ( ( A - B ) / 2 ) e. RR )
8	4, 7	readdcld 8073	. . 3 |- ( ph -> ( ( F ` K ) + ( ( A - B ) / 2 ) ) e. RR )
9		recvguniqlem.lt1	. . 3 |- ( ph -> A < ( ( F ` K ) + ( ( A - B ) / 2 ) ) )
10	5, 7	readdcld 8073	. . . . 5 |- ( ph -> ( B + ( ( A - B ) / 2 ) ) e. RR )
11		recvguniqlem.lt2	. . . . 5 |- ( ph -> ( F ` K ) < ( B + ( ( A - B ) / 2 ) ) )
12	4, 10, 7, 11	ltadd1dd 8600	. . . 4 |- ( ph -> ( ( F ` K ) + ( ( A - B ) / 2 ) ) < ( ( B + ( ( A - B ) / 2 ) ) + ( ( A - B ) / 2 ) ) )
13	5	recnd 8072	. . . . . 6 |- ( ph -> B e. CC )
14	7	recnd 8072	. . . . . 6 |- ( ph -> ( ( A - B ) / 2 ) e. CC )
15	13, 14, 14	addassd 8066	. . . . 5 |- ( ph -> ( ( B + ( ( A - B ) / 2 ) ) + ( ( A - B ) / 2 ) ) = ( B + ( ( ( A - B ) / 2 ) + ( ( A - B ) / 2 ) ) ) )
16	6	recnd 8072	. . . . . . 7 |- ( ph -> ( A - B ) e. CC )
17	16	2halvesd 9254	. . . . . 6 |- ( ph -> ( ( ( A - B ) / 2 ) + ( ( A - B ) / 2 ) ) = ( A - B ) )
18	17	oveq2d 5941	. . . . 5 |- ( ph -> ( B + ( ( ( A - B ) / 2 ) + ( ( A - B ) / 2 ) ) ) = ( B + ( A - B ) ) )
19	1	recnd 8072	. . . . . 6 |- ( ph -> A e. CC )
20	13, 19	pncan3d 8357	. . . . 5 |- ( ph -> ( B + ( A - B ) ) = A )
21	15, 18, 20	3eqtrd 2233	. . . 4 |- ( ph -> ( ( B + ( ( A - B ) / 2 ) ) + ( ( A - B ) / 2 ) ) = A )
22	12, 21	breqtrd 4060	. . 3 |- ( ph -> ( ( F ` K ) + ( ( A - B ) / 2 ) ) < A )
23	1, 8, 1, 9, 22	lttrd 8169	. 2 |- ( ph -> A < A )
24	1	ltnrd 8155	. 2 |- ( ph -> -. A < A )
25	23, 24	pm2.21fal 1384	1 |- ( ph -> F. )

Syntax hints:   -> wi 4   F. wfal 1369   e. wcel 2167   class class class wbr 4034   -->wf 5255   ` cfv 5259  (class class class)co 5925   RRcr 7895   + caddc 7899   < clt 8078   - cmin 8214   / cdiv 8716   NNcn 9007   2c2 9058

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-po 4332  df-iso 4333  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-2 9066

This theorem is referenced by:  recvguniq  11177