Theorem recvguniqlem 11305

Description: Lemma for recvguniq 11306. 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 5716	. . . 4 |- ( ph -> ( F ` K ) e. RR )
5		recvguniqlem.b	. . . . . 6 |- ( ph -> B e. RR )
6	1, 5	resubcld 8453	. . . . 5 |- ( ph -> ( A - B ) e. RR )
7	6	rehalfcld 9284	. . . 4 |- ( ph -> ( ( A - B ) / 2 ) e. RR )
8	4, 7	readdcld 8102	. . 3 |- ( ph -> ( ( F ` K ) + ( ( A - B ) / 2 ) ) e. RR )
9		recvguniqlem.lt1	. . 3 |- ( ph -> A < ( ( F ` K ) + ( ( A - B ) / 2 ) ) )
10	5, 7	readdcld 8102	. . . . 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 8629	. . . 4 |- ( ph -> ( ( F ` K ) + ( ( A - B ) / 2 ) ) < ( ( B + ( ( A - B ) / 2 ) ) + ( ( A - B ) / 2 ) ) )
13	5	recnd 8101	. . . . . 6 |- ( ph -> B e. CC )
14	7	recnd 8101	. . . . . 6 |- ( ph -> ( ( A - B ) / 2 ) e. CC )
15	13, 14, 14	addassd 8095	. . . . 5 |- ( ph -> ( ( B + ( ( A - B ) / 2 ) ) + ( ( A - B ) / 2 ) ) = ( B + ( ( ( A - B ) / 2 ) + ( ( A - B ) / 2 ) ) ) )
16	6	recnd 8101	. . . . . . 7 |- ( ph -> ( A - B ) e. CC )
17	16	2halvesd 9283	. . . . . 6 |- ( ph -> ( ( ( A - B ) / 2 ) + ( ( A - B ) / 2 ) ) = ( A - B ) )
18	17	oveq2d 5960	. . . . 5 |- ( ph -> ( B + ( ( ( A - B ) / 2 ) + ( ( A - B ) / 2 ) ) ) = ( B + ( A - B ) ) )
19	1	recnd 8101	. . . . . 6 |- ( ph -> A e. CC )
20	13, 19	pncan3d 8386	. . . . 5 |- ( ph -> ( B + ( A - B ) ) = A )
21	15, 18, 20	3eqtrd 2242	. . . 4 |- ( ph -> ( ( B + ( ( A - B ) / 2 ) ) + ( ( A - B ) / 2 ) ) = A )
22	12, 21	breqtrd 4070	. . 3 |- ( ph -> ( ( F ` K ) + ( ( A - B ) / 2 ) ) < A )
23	1, 8, 1, 9, 22	lttrd 8198	. 2 |- ( ph -> A < A )
24	1	ltnrd 8184	. 2 |- ( ph -> -. A < A )
25	23, 24	pm2.21fal 1393	1 |- ( ph -> F. )

Syntax hints:   -> wi 4   F. wfal 1378   e. wcel 2176   class class class wbr 4044   -->wf 5267   ` cfv 5271  (class class class)co 5944   RRcr 7924   + caddc 7928   < clt 8107   - cmin 8243   / cdiv 8745   NNcn 9036   2c2 9087

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-po 4343  df-iso 4344  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-2 9095

This theorem is referenced by:  recvguniq  11306