Theorem fzmmmeqm 10293

Description: Subtracting the difference of a member of a finite range of integers and the lower bound of the range from the difference of the upper bound and the lower bound of the range results in the difference of the upper bound of the range and the member. (Contributed by Alexander van der Vekens, 27-May-2018.)

Assertion

Ref	Expression
fzmmmeqm	|- ( M e. ( L ... N ) -> ( ( N - L ) - ( M - L ) ) = ( N - M ) )

Proof of Theorem fzmmmeqm

Step	Hyp	Ref	Expression
1		elfz2 10250	. . 3 |- ( M e. ( L ... N ) <-> ( ( L e. ZZ /\ N e. ZZ /\ M e. ZZ ) /\ ( L <_ M /\ M <_ N ) ) )
2		zcn 9484	. . . . . 6 |- ( N e. ZZ -> N e. CC )
3		zcn 9484	. . . . . 6 |- ( M e. ZZ -> M e. CC )
4		zcn 9484	. . . . . 6 |- ( L e. ZZ -> L e. CC )
5	2, 3, 4	3anim123i 1210	. . . . 5 |- ( ( N e. ZZ /\ M e. ZZ /\ L e. ZZ ) -> ( N e. CC /\ M e. CC /\ L e. CC ) )
6	5	3comr 1237	. . . 4 |- ( ( L e. ZZ /\ N e. ZZ /\ M e. ZZ ) -> ( N e. CC /\ M e. CC /\ L e. CC ) )
7	6	adantr 276	. . 3 |- ( ( ( L e. ZZ /\ N e. ZZ /\ M e. ZZ ) /\ ( L <_ M /\ M <_ N ) ) -> ( N e. CC /\ M e. CC /\ L e. CC ) )
8	1, 7	sylbi 121	. 2 |- ( M e. ( L ... N ) -> ( N e. CC /\ M e. CC /\ L e. CC ) )
9		nnncan2 8416	. 2 |- ( ( N e. CC /\ M e. CC /\ L e. CC ) -> ( ( N - L ) - ( M - L ) ) = ( N - M ) )
10	8, 9	syl 14	1 |- ( M e. ( L ... N ) -> ( ( N - L ) - ( M - L ) ) = ( N - M ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   class class class wbr 4088  (class class class)co 6018   CCcc 8030   <_ cle 8215   - cmin 8350   ZZcz 9479   ...cfz 10243

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-sub 8352  df-neg 8353  df-z 9480  df-fz 10244

This theorem is referenced by:  swrdccatin2  11310