2lgslem1a - Intuitionistic Logic Explorer

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Theorem 2lgslem1a 15732

Description: Lemma 1 for 2lgslem1 15735. (Contributed by AV, 18-Jun-2021.)

**Assertion**
Ref	Expression
2lgslem1a	$/\$ $/\$

Distinct variable group:

Proof of Theorem 2lgslem1a Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prmnn 12598	. . . . . . . . . 10
2	1	nnnn0d 9390	. . . . . . . . 9
3	2	ad2antrr 488	. . . . . . . 8 $/\$ $/\$
4		4nn 9242	. . . . . . . 8
5	3, 4	jctir 313	. . . . . . 7 $/\$ $/\$ $/\$
6		fldivnn0 10482	. . . . . . 7 $/\$
7		nn0p1nn 9376	. . . . . . 7
8	5, 6, 7	3syl 17	. . . . . 6 $/\$ $/\$
9		elnnuz 9727	. . . . . 6
10	8, 9	sylib 122	. . . . 5 $/\$ $/\$
11		fzss1 10227	. . . . 5
12		rexss 3271	. . . . 5 $/\$
13	10, 11, 12	3syl 17	. . . 4 $/\$ $/\$ $/\$
14		ancom 266	. . . . . 6 $/\$ $/\$
15	2, 4	jctir 313	. . . . . . . . . . . . . . . . 17 $/\$
16	15, 6	syl 14	. . . . . . . . . . . . . . . 16
17	16	nn0zd 9535	. . . . . . . . . . . . . . 15
18	17	ad2antrr 488	. . . . . . . . . . . . . 14 $/\$ $/\$
19		elfzelz 10189	. . . . . . . . . . . . . 14
20		zltp1le 9469	. . . . . . . . . . . . . 14 $/\$
21	18, 19, 20	syl2an 289	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
22	21	bicomd 141	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
23	22	anbi1d 465	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
24	19	adantl 277	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
25	17	peano2zd 9540	. . . . . . . . . . . . . 14
26	25	adantr 276	. . . . . . . . . . . . 13 $/\$
27	26	ad2antrr 488	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
28		prmz 12599	. . . . . . . . . . . . . . 15
29		oddm1d2 12369	. . . . . . . . . . . . . . 15
30	28, 29	syl 14	. . . . . . . . . . . . . 14
31	30	biimpa 296	. . . . . . . . . . . . 13 $/\$
32	31	ad2antrr 488	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
33		elfz 10178	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
34	24, 27, 32, 33	syl3anc 1252	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
35		elfzle2 10192	. . . . . . . . . . . . 13
36	35	adantl 277	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
37	36	biantrud 304	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
38	23, 34, 37	3bitr4d 220	. . . . . . . . . 10 $/\$ $/\$ $/\$
39	28	ad2antrr 488	. . . . . . . . . . 11 $/\$ $/\$
40		2lgslem1a2 15731	. . . . . . . . . . 11 $/\$
41	39, 19, 40	syl2an 289	. . . . . . . . . 10 $/\$ $/\$ $/\$
42	38, 41	bitrd 188	. . . . . . . . 9 $/\$ $/\$ $/\$
43		2lgslem1a1 15730	. . . . . . . . . . . . 13 $/\$
44	1, 43	sylan 283	. . . . . . . . . . . 12 $/\$
45	44	adantr 276	. . . . . . . . . . 11 $/\$ $/\$
46		oveq1 5981	. . . . . . . . . . . . 13
47	46	oveq1d 5989	. . . . . . . . . . . . 13
48	46, 47	eqeq12d 2224	. . . . . . . . . . . 12
49	48	rspccva 2886	. . . . . . . . . . 11 $/\$
50	45, 49	sylan 283	. . . . . . . . . 10 $/\$ $/\$ $/\$
51	50	breq2d 4074	. . . . . . . . 9 $/\$ $/\$ $/\$
52	42, 51	bitrd 188	. . . . . . . 8 $/\$ $/\$ $/\$
53		oveq1 5981	. . . . . . . . . 10
54	53	eqcomd 2215	. . . . . . . . 9
55	54	breq2d 4074	. . . . . . . 8
56	52, 55	sylan9bb 462	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
57	56	pm5.32da 452	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
58	14, 57	bitrid 192	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
59	58	rexbidva 2507	. . . 4 $/\$ $/\$ $/\$ $/\$
60	13, 59	bitrd 188	. . 3 $/\$ $/\$ $/\$
61	60	bicomd 141	. 2 $/\$ $/\$ $/\$
62	61	rabbidva 2767	1 $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105

wceq 1375

wcel 2180

wral 2488

wrex 2489

crab 2492

wss 3177 class class class wbr 4062

cfv 5294 (class class class)co 5974

c1 7968

caddc 7970

cmul 7972

clt 8149

cle 8150

cmin 8285

cdiv 8787

cn 9078

c2 9129

c4 9131

cn0 9337

cz 9414

cuz 9690

cfz 10172

cfl 10455

cmo 10511

cdvds 12264

cprime 12595

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 617 ax-in2 618 ax-io 713 ax-5 1473 ax-7 1474 ax-gen 1475 ax-ie1 1519 ax-ie2 1520 ax-8 1530 ax-10 1531 ax-11 1532 ax-i12 1533 ax-bndl 1535 ax-4 1536 ax-17 1552 ax-i9 1556 ax-ial 1560 ax-i5r 1561 ax-13 2182 ax-14 2183 ax-ext 2191 ax-sep 4181 ax-pow 4237 ax-pr 4272 ax-un 4501 ax-setind 4606 ax-cnex 8058 ax-resscn 8059 ax-1cn 8060 ax-1re 8061 ax-icn 8062 ax-addcl 8063 ax-addrcl 8064 ax-mulcl 8065 ax-mulrcl 8066 ax-addcom 8067 ax-mulcom 8068 ax-addass 8069 ax-mulass 8070 ax-distr 8071 ax-i2m1 8072 ax-0lt1 8073 ax-1rid 8074 ax-0id 8075 ax-rnegex 8076 ax-precex 8077 ax-cnre 8078 ax-pre-ltirr 8079 ax-pre-ltwlin 8080 ax-pre-lttrn 8081 ax-pre-apti 8082 ax-pre-ltadd 8083 ax-pre-mulgt0 8084 ax-pre-mulext 8085 ax-arch 8086

This theorem depends on definitions: df-bi 117 df-3or 984 df-3an 985 df-tru 1378 df-fal 1381 df-xor 1398 df-nf 1487 df-sb 1789 df-eu 2060 df-mo 2061 df-clab 2196 df-cleq 2202 df-clel 2205 df-nfc 2341 df-ne 2381 df-nel 2476 df-ral 2493 df-rex 2494 df-reu 2495 df-rmo 2496 df-rab 2497 df-v 2781 df-sbc 3009 df-csb 3105 df-dif 3179 df-un 3181 df-in 3183 df-ss 3190 df-pw 3631 df-sn 3652 df-pr 3653 df-op 3655 df-uni 3868 df-int 3903 df-iun 3946 df-br 4063 df-opab 4125 df-mpt 4126 df-id 4361 df-po 4364 df-iso 4365 df-xp 4702 df-rel 4703 df-cnv 4704 df-co 4705 df-dm 4706 df-rn 4707 df-res 4708 df-ima 4709 df-iota 5254 df-fun 5296 df-fn 5297 df-f 5298 df-fv 5302 df-riota 5927 df-ov 5977 df-oprab 5978 df-mpo 5979 df-1st 6256 df-2nd 6257 df-pnf 8151 df-mnf 8152 df-xr 8153 df-ltxr 8154 df-le 8155 df-sub 8287 df-neg 8288 df-reap 8690 df-ap 8697 df-div 8788 df-inn 9079 df-2 9137 df-3 9138 df-4 9139 df-n0 9338 df-z 9415 df-uz 9691 df-q 9783 df-rp 9818 df-fz 10173 df-fl 10457 df-mod 10512 df-dvds 12265 df-prm 12596

This theorem is referenced by: 2lgslem1 15735

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