2lgslem1a - Intuitionistic Logic Explorer

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Theorem 2lgslem1a 16010

Description: Lemma 1 for 2lgslem1 16013. (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 12815	. . . . . . . . . 10
2	1	nnnn0d 9558	. . . . . . . . 9
3	2	ad2antrr 488	. . . . . . . 8 $/\$ $/\$
4		4nn 9406	. . . . . . . 8
5	3, 4	jctir 313	. . . . . . 7 $/\$ $/\$ $/\$
6		fldivnn0 10662	. . . . . . 7 $/\$
7		nn0p1nn 9540	. . . . . . 7
8	5, 6, 7	3syl 17	. . . . . 6 $/\$ $/\$
9		elnnuz 9897	. . . . . 6
10	8, 9	sylib 122	. . . . 5 $/\$ $/\$
11		fzss1 10403	. . . . 5
12		rexss 3307	. . . . 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 9704	. . . . . . . . . . . . . . 15
18	17	ad2antrr 488	. . . . . . . . . . . . . 14 $/\$ $/\$
19		elfzelz 10365	. . . . . . . . . . . . . 14
20		zltp1le 9637	. . . . . . . . . . . . . 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 9709	. . . . . . . . . . . . . 14
26	25	adantr 276	. . . . . . . . . . . . 13 $/\$
27	26	ad2antrr 488	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
28		prmz 12816	. . . . . . . . . . . . . . 15
29		oddm1d2 12586	. . . . . . . . . . . . . . 15
30	28, 29	syl 14	. . . . . . . . . . . . . 14
31	30	biimpa 296	. . . . . . . . . . . . 13 $/\$
32	31	ad2antrr 488	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
33		elfz 10354	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
34	24, 27, 32, 33	syl3anc 1274	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
35		elfzle2 10368	. . . . . . . . . . . . 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 16009	. . . . . . . . . . 11 $/\$
41	39, 19, 40	syl2an 289	. . . . . . . . . 10 $/\$ $/\$ $/\$
42	38, 41	bitrd 188	. . . . . . . . 9 $/\$ $/\$ $/\$
43		2lgslem1a1 16008	. . . . . . . . . . . . 13 $/\$
44	1, 43	sylan 283	. . . . . . . . . . . 12 $/\$
45	44	adantr 276	. . . . . . . . . . 11 $/\$ $/\$
46		oveq1 6059	. . . . . . . . . . . . 13
47	46	oveq1d 6067	. . . . . . . . . . . . 13
48	46, 47	eqeq12d 2249	. . . . . . . . . . . 12
49	48	rspccva 2922	. . . . . . . . . . 11 $/\$
50	45, 49	sylan 283	. . . . . . . . . 10 $/\$ $/\$ $/\$
51	50	breq2d 4123	. . . . . . . . 9 $/\$ $/\$ $/\$
52	42, 51	bitrd 188	. . . . . . . 8 $/\$ $/\$ $/\$
53		oveq1 6059	. . . . . . . . . 10
54	53	eqcomd 2240	. . . . . . . . 9
55	54	breq2d 4123	. . . . . . . 8
56	52, 55	sylan9bb 462	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
57	56	pm5.32da 452	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
58	14, 57	bitrid 192	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
59	58	rexbidva 2541	. . . 4 $/\$ $/\$ $/\$ $/\$
60	13, 59	bitrd 188	. . 3 $/\$ $/\$ $/\$
61	60	bicomd 141	. 2 $/\$ $/\$ $/\$
62	61	rabbidva 2803	1 $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105

wceq 1398

wcel 2205

wral 2522

wrex 2523

crab 2526

wss 3213 class class class wbr 4111

cfv 5354 (class class class)co 6052

c1 8133

caddc 8135

cmul 8137

clt 8313

cle 8314

cmin 8449

cdiv 8951

cn 9242

c2 9293

c4 9295

cn0 9501

cz 9582

cuz 9859

cfz 10348

cfl 10635

cmo 10691

cdvds 12481

cprime 12812

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-sep 4230 ax-pow 4289 ax-pr 4324 ax-un 4556 ax-setind 4661 ax-cnex 8223 ax-resscn 8224 ax-1cn 8225 ax-1re 8226 ax-icn 8227 ax-addcl 8228 ax-addrcl 8229 ax-mulcl 8230 ax-mulrcl 8231 ax-addcom 8232 ax-mulcom 8233 ax-addass 8234 ax-mulass 8235 ax-distr 8236 ax-i2m1 8237 ax-0lt1 8238 ax-1rid 8239 ax-0id 8240 ax-rnegex 8241 ax-precex 8242 ax-cnre 8243 ax-pre-ltirr 8244 ax-pre-ltwlin 8245 ax-pre-lttrn 8246 ax-pre-apti 8247 ax-pre-ltadd 8248 ax-pre-mulgt0 8249 ax-pre-mulext 8250 ax-arch 8251

This theorem depends on definitions: df-bi 117 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-xor 1421 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3045 df-csb 3141 df-dif 3215 df-un 3217 df-in 3219 df-ss 3226 df-pw 3673 df-sn 3697 df-pr 3698 df-op 3700 df-uni 3917 df-int 3952 df-iun 3995 df-br 4112 df-opab 4174 df-mpt 4175 df-id 4416 df-po 4419 df-iso 4420 df-xp 4757 df-rel 4758 df-cnv 4759 df-co 4760 df-dm 4761 df-rn 4762 df-res 4763 df-ima 4764 df-iota 5314 df-fun 5356 df-fn 5357 df-f 5358 df-fv 5362 df-riota 6005 df-ov 6055 df-oprab 6056 df-mpo 6057 df-1st 6336 df-2nd 6337 df-pnf 8315 df-mnf 8316 df-xr 8317 df-ltxr 8318 df-le 8319 df-sub 8451 df-neg 8452 df-reap 8854 df-ap 8861 df-div 8952 df-inn 9243 df-2 9301 df-3 9302 df-4 9303 df-n0 9502 df-z 9583 df-uz 9860 df-q 9958 df-rp 9993 df-fz 10349 df-fl 10637 df-mod 10692 df-dvds 12482 df-prm 12813

This theorem is referenced by: 2lgslem1 16013

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