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Theorem ax5seglem3a 25810
Description: Lemma for ax5seg 25818. (Contributed by Scott Fenton, 7-May-2015.)
Assertion
Ref Expression
ax5seglem3a |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> ( ( ( A ` j ) - ( C ` j ) ) e. RR /\ ( ( D ` j ) - ( F ` j ) ) e. RR ) )
Proof of Theorem ax5seglem3a
StepHypRef Expression
1 simpl21 1139 . . . 4 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> A e. ( EE ` N ) )
2 fveere 25781 . . . 4 |- ( ( A e. ( EE ` N ) /\ j e. ( 1 ... N ) ) -> ( A ` j ) e. RR )
31, 2sylancom 701 . . 3 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> ( A ` j ) e. RR )
4 simpl23 1141 . . . 4 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> C e. ( EE ` N ) )
5 fveere 25781 . . . 4 |- ( ( C e. ( EE ` N ) /\ j e. ( 1 ... N ) ) -> ( C ` j ) e. RR )
64, 5sylancom 701 . . 3 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> ( C ` j ) e. RR )
73, 6resubcld 10458 . 2 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> ( ( A ` j ) - ( C ` j ) ) e. RR )
8 simpl31 1142 . . . 4 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> D e. ( EE ` N ) )
9 fveere 25781 . . . 4 |- ( ( D e. ( EE ` N ) /\ j e. ( 1 ... N ) ) -> ( D ` j ) e. RR )
108, 9sylancom 701 . . 3 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> ( D ` j ) e. RR )
11 simpl33 1144 . . . 4 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> F e. ( EE ` N ) )
12 fveere 25781 . . . 4 |- ( ( F e. ( EE ` N ) /\ j e. ( 1 ... N ) ) -> ( F ` j ) e. RR )
1311, 12sylancom 701 . . 3 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> ( F ` j ) e. RR )
1410, 13resubcld 10458 . 2 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> ( ( D ` j ) - ( F ` j ) ) e. RR )
157, 14jca 554 1 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> ( ( ( A ` j ) - ( C ` j ) ) e. RR /\ ( ( D ` j ) - ( F ` j ) ) e. RR ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   ` cfv 5888  (class class class)co 6650   RRcr 9935   1c1 9937   - cmin 10266   NNcn 11020   ...cfz 12326   EEcee 25768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-sub 10268  df-neg 10269  df-ee 25771
This theorem is referenced by:  ax5seglem3  25811
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