Theorem fz1eqblem 13608

Description: Lemma for fz1eqb 13609 $.

Hypotheses

Ref	Expression
fz1eqblem.1	|- M e. NN0
fz1eqblem.2	|- N e. NN0

Assertion

Ref	Expression
fz1eqblem	|- ((1...M) = (1...N) <-> M = N)

Proof of Theorem fz1eqblem

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . 4 |- (M = 0 -> (1...M) = (1...0))
2		opreq2 4890	. . . . 5 |- (N = 0 -> (1...N) = (1...0))
3	2	eqcomd 1889	. . . 4 |- (N = 0 -> (1...0) = (1...N))
4	1, 3	sylan9eq 1948	. . 3 |- ((M = 0 /\ N = 0) -> (1...M) = (1...N))
5		eqtr3 1907	. . 3 |- ((M = 0 /\ N = 0) -> M = N)
6		pm5.1 740	. . 3 |- (((1...M) = (1...N) /\ M = N) -> ((1...M) = (1...N) <-> M = N))
7	4, 5, 6	syl11anc 524	. 2 |- ((M = 0 /\ N = 0) -> ((1...M) = (1...N) <-> M = N))
8		fz1eqblem.1	. . . . . . . . . 10 |- M e. NN0
9		fz1n 13607	. . . . . . . . . 10 |- (M e. NN0 -> ((1...M) = (/) <-> M = 0))
10	8, 9	ax-mp 7	. . . . . . . . 9 |- ((1...M) = (/) <-> M = 0)
11	10	biimpri 169	. . . . . . . 8 |- (M = 0 -> (1...M) = (/))
12	11	eqeq1d 1892	. . . . . . 7 |- (M = 0 -> ((1...M) = (1...N) <-> (/) = (1...N)))
13		fz1eqblem.2	. . . . . . . . . 10 |- N e. NN0
14		fz1n 13607	. . . . . . . . . 10 |- (N e. NN0 -> ((1...N) = (/) <-> N = 0))
15	13, 14	ax-mp 7	. . . . . . . . 9 |- ((1...N) = (/) <-> N = 0)
16	15	biimpi 168	. . . . . . . 8 |- ((1...N) = (/) -> N = 0)
17	16	eqcoms 1887	. . . . . . 7 |- ((/) = (1...N) -> N = 0)
18	12, 17	syl6bi 231	. . . . . 6 |- (M = 0 -> ((1...M) = (1...N) -> N = 0))
19	18	con3d 111	. . . . 5 |- (M = 0 -> (-. N = 0 -> -. (1...M) = (1...N)))
20	19	imp 377	. . . 4 |- ((M = 0 /\ -. N = 0) -> -. (1...M) = (1...N))
21		eqeq1 1890	. . . . . . 7 |- (M = 0 -> (M = N <-> 0 = N))
22		eqcom 1886	. . . . . . 7 |- (0 = N <-> N = 0)
23	21, 22	syl6bb 595	. . . . . 6 |- (M = 0 -> (M = N <-> N = 0))
24	23	notbid 673	. . . . 5 |- (M = 0 -> (-. M = N <-> -. N = 0))
25	24	biimpar 461	. . . 4 |- ((M = 0 /\ -. N = 0) -> -. M = N)
26		pm5.1 740	. . . 4 |- ((-. (1...M) = (1...N) /\ -. M = N) -> (-. (1...M) = (1...N) <-> -. M = N))
27	20, 25, 26	syl11anc 524	. . 3 |- ((M = 0 /\ -. N = 0) -> (-. (1...M) = (1...N) <-> -. M = N))
28	27	con4bid 583	. 2 |- ((M = 0 /\ -. N = 0) -> ((1...M) = (1...N) <-> M = N))
29	15	biimpri 169	. . . . . . . 8 |- (N = 0 -> (1...N) = (/))
30	29	eqeq2d 1895	. . . . . . 7 |- (N = 0 -> ((1...M) = (1...N) <-> (1...M) = (/)))
31	10	biimpi 168	. . . . . . 7 |- ((1...M) = (/) -> M = 0)
32	30, 31	syl6bi 231	. . . . . 6 |- (N = 0 -> ((1...M) = (1...N) -> M = 0))
33	32	con3d 111	. . . . 5 |- (N = 0 -> (-. M = 0 -> -. (1...M) = (1...N)))
34	33	impcom 378	. . . 4 |- ((-. M = 0 /\ N = 0) -> -. (1...M) = (1...N))
35		eqeq2 1893	. . . . . 6 |- (N = 0 -> (M = N <-> M = 0))
36	35	notbid 673	. . . . 5 |- (N = 0 -> (-. M = N <-> -. M = 0))
37	36	biimparc 463	. . . 4 |- ((-. M = 0 /\ N = 0) -> -. M = N)
38	34, 37, 26	syl11anc 524	. . 3 |- ((-. M = 0 /\ N = 0) -> (-. (1...M) = (1...N) <-> -. M = N))
39	38	con4bid 583	. 2 |- ((-. M = 0 /\ N = 0) -> ((1...M) = (1...N) <-> M = N))
40		fzopth 7674	. . . . 5 |- ((M e. (ZZ>=` 1) /\ N e. (ZZ>=` 1)) -> ((1...M) = (1...N) <-> (1 = 1 /\ M = N)))
41		elnnuz 7609	. . . . 5 |- (M e. NN <-> M e. (ZZ>=` 1))
42		elnnuz 7609	. . . . 5 |- (N e. NN <-> N e. (ZZ>=` 1))
43	40, 41, 42	syl2anb 504	. . . 4 |- ((M e. NN /\ N e. NN) -> ((1...M) = (1...N) <-> (1 = 1 /\ M = N)))
44		eqid 1884	. . . . 5 |- 1 = 1
45	44	biantrur 794	. . . 4 |- (M = N <-> (1 = 1 /\ M = N))
46	43, 45	syl6bbr 597	. . 3 |- ((M e. NN /\ N e. NN) -> ((1...M) = (1...N) <-> M = N))
47		elnn00nn 13602	. . . . 5 |- (M e. NN0 <-> (M = 0 \/ M e. NN))
48	8, 47	mpbi 206	. . . 4 |- (M = 0 \/ M e. NN)
49	48	ori 247	. . 3 |- (-. M = 0 -> M e. NN)
50		elnn00nn 13602	. . . . 5 |- (N e. NN0 <-> (N = 0 \/ N e. NN))
51	13, 50	mpbi 206	. . . 4 |- (N = 0 \/ N e. NN)
52	51	ori 247	. . 3 |- (-. N = 0 -> N e. NN)
53	46, 49, 52	syl2an 503	. 2 |- ((-. M = 0 /\ -. N = 0) -> ((1...M) = (1...N) <-> M = N))
54	7, 28, 39, 53	4cases 832	1 |- ((1...M) = (1...N) <-> M = N)

Syntax hints:  -. wn 2   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  (/)c0 2875  ` cfv 3998  (class class class)co 4884  0cc0 6386  1c1 6387  NNcn 6449  NN0cn0 6450  ZZ>=cuz 7586  ...cfz 7637

This theorem is referenced by:  fz1eqb 13609

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-n 7108  df-n0 7309  df-z 7345  df-uz 7587  df-fz 7638

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