Theorem recvguniqlem 10958

Description: Lemma for recvguniq 10959. 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	ffvelrnd 5632	. . . 4 |- ( ph -> ( F ` K ) e. RR )
5		recvguniqlem.b	. . . . . 6 |- ( ph -> B e. RR )
6	1, 5	resubcld 8300	. . . . 5 |- ( ph -> ( A - B ) e. RR )
7	6	rehalfcld 9124	. . . 4 |- ( ph -> ( ( A - B ) / 2 ) e. RR )
8	4, 7	readdcld 7949	. . 3 |- ( ph -> ( ( F ` K ) + ( ( A - B ) / 2 ) ) e. RR )
9		recvguniqlem.lt1	. . 3 |- ( ph -> A < ( ( F ` K ) + ( ( A - B ) / 2 ) ) )
10	5, 7	readdcld 7949	. . . . 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 8475	. . . 4 |- ( ph -> ( ( F ` K ) + ( ( A - B ) / 2 ) ) < ( ( B + ( ( A - B ) / 2 ) ) + ( ( A - B ) / 2 ) ) )
13	5	recnd 7948	. . . . . 6 |- ( ph -> B e. CC )
14	7	recnd 7948	. . . . . 6 |- ( ph -> ( ( A - B ) / 2 ) e. CC )
15	13, 14, 14	addassd 7942	. . . . 5 |- ( ph -> ( ( B + ( ( A - B ) / 2 ) ) + ( ( A - B ) / 2 ) ) = ( B + ( ( ( A - B ) / 2 ) + ( ( A - B ) / 2 ) ) ) )
16	6	recnd 7948	. . . . . . 7 |- ( ph -> ( A - B ) e. CC )
17	16	2halvesd 9123	. . . . . 6 |- ( ph -> ( ( ( A - B ) / 2 ) + ( ( A - B ) / 2 ) ) = ( A - B ) )
18	17	oveq2d 5869	. . . . 5 |- ( ph -> ( B + ( ( ( A - B ) / 2 ) + ( ( A - B ) / 2 ) ) ) = ( B + ( A - B ) ) )
19	1	recnd 7948	. . . . . 6 |- ( ph -> A e. CC )
20	13, 19	pncan3d 8233	. . . . 5 |- ( ph -> ( B + ( A - B ) ) = A )
21	15, 18, 20	3eqtrd 2207	. . . 4 |- ( ph -> ( ( B + ( ( A - B ) / 2 ) ) + ( ( A - B ) / 2 ) ) = A )
22	12, 21	breqtrd 4015	. . 3 |- ( ph -> ( ( F ` K ) + ( ( A - B ) / 2 ) ) < A )
23	1, 8, 1, 9, 22	lttrd 8045	. 2 |- ( ph -> A < A )
24	1	ltnrd 8031	. 2 |- ( ph -> -. A < A )
25	23, 24	pm2.21fal 1368	1 |- ( ph -> F. )

Syntax hints:   -> wi 4   F. wfal 1353   e. wcel 2141   class class class wbr 3989   -->wf 5194   ` cfv 5198  (class class class)co 5853   RRcr 7773   + caddc 7777   < clt 7954   - cmin 8090   / cdiv 8589   NNcn 8878   2c2 8929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-2 8937

This theorem is referenced by:  recvguniq  10959