2lgslem1a - Intuitionistic Logic Explorer

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Theorem 2lgslem1a 16090

Description: Lemma 1 for 2lgslem1 16093. (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 12835	. . . . . . . . . 10
2	1	nnnn0d 9573	. . . . . . . . 9
3	2	ad2antrr 488	. . . . . . . 8 $/\$ $/\$
4		4nn 9421	. . . . . . . 8
5	3, 4	jctir 313	. . . . . . 7 $/\$ $/\$ $/\$
6		fldivnn0 10682	. . . . . . 7 $/\$
7		nn0p1nn 9555	. . . . . . 7
8	5, 6, 7	3syl 17	. . . . . 6 $/\$ $/\$
9		elnnuz 9912	. . . . . 6
10	8, 9	sylib 122	. . . . 5 $/\$ $/\$
11		fzss1 10421	. . . . 5
12		rexss 3309	. . . . 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 9719	. . . . . . . . . . . . . . 15
18	17	ad2antrr 488	. . . . . . . . . . . . . 14 $/\$ $/\$
19		elfzelz 10381	. . . . . . . . . . . . . 14
20		zltp1le 9652	. . . . . . . . . . . . . 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 9724	. . . . . . . . . . . . . 14
26	25	adantr 276	. . . . . . . . . . . . 13 $/\$
27	26	ad2antrr 488	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
28		prmz 12836	. . . . . . . . . . . . . . 15
29		oddm1d2 12606	. . . . . . . . . . . . . . 15
30	28, 29	syl 14	. . . . . . . . . . . . . 14
31	30	biimpa 296	. . . . . . . . . . . . 13 $/\$
32	31	ad2antrr 488	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
33		elfz 10370	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
34	24, 27, 32, 33	syl3anc 1274	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
35		elfzle2 10385	. . . . . . . . . . . . 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 16089	. . . . . . . . . . 11 $/\$
41	39, 19, 40	syl2an 289	. . . . . . . . . 10 $/\$ $/\$ $/\$
42	38, 41	bitrd 188	. . . . . . . . 9 $/\$ $/\$ $/\$
43		2lgslem1a1 16088	. . . . . . . . . . . . 13 $/\$
44	1, 43	sylan 283	. . . . . . . . . . . 12 $/\$
45	44	adantr 276	. . . . . . . . . . 11 $/\$ $/\$
46		oveq1 6065	. . . . . . . . . . . . 13
47	46	oveq1d 6073	. . . . . . . . . . . . 13
48	46, 47	eqeq12d 2249	. . . . . . . . . . . 12
49	48	rspccva 2922	. . . . . . . . . . 11 $/\$
50	45, 49	sylan 283	. . . . . . . . . 10 $/\$ $/\$ $/\$
51	50	breq2d 4126	. . . . . . . . 9 $/\$ $/\$ $/\$
52	42, 51	bitrd 188	. . . . . . . 8 $/\$ $/\$ $/\$
53		oveq1 6065	. . . . . . . . . 10
54	53	eqcomd 2240	. . . . . . . . 9
55	54	breq2d 4126	. . . . . . . 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 3214 class class class wbr 4114

cfv 5357 (class class class)co 6058

c1 8144

caddc 8146

cmul 8148

clt 8324

cle 8325

cmin 8461

cdiv 8966

cn 9257

c2 9308

c4 9310

cn0 9516

cz 9597

cuz 9874

cfz 10364

cfl 10655

cmo 10711

cdvds 12501

cprime 12832

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 4233 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262

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 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-id 4419 df-po 4422 df-iso 4423 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8463 df-neg 8464 df-reap 8867 df-ap 8874 df-div 8967 df-inn 9258 df-2 9316 df-3 9317 df-4 9318 df-n0 9517 df-z 9598 df-uz 9875 df-q 9973 df-rp 10008 df-fz 10365 df-fl 10657 df-mod 10712 df-dvds 12502 df-prm 12833

This theorem is referenced by: 2lgslem1 16093

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