2lgslem1a - Intuitionistic Logic Explorer

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Theorem 2lgslem1a 15236

Description: Lemma 1 for 2lgslem1 15239. (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 12251	. . . . . . . . . 10
2	1	nnnn0d 9296	. . . . . . . . 9
3	2	ad2antrr 488	. . . . . . . 8 $/\$ $/\$
4		4nn 9148	. . . . . . . 8
5	3, 4	jctir 313	. . . . . . 7 $/\$ $/\$ $/\$
6		fldivnn0 10367	. . . . . . 7 $/\$
7		nn0p1nn 9282	. . . . . . 7
8	5, 6, 7	3syl 17	. . . . . 6 $/\$ $/\$
9		elnnuz 9632	. . . . . 6
10	8, 9	sylib 122	. . . . 5 $/\$ $/\$
11		fzss1 10132	. . . . 5
12		rexss 3247	. . . . 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 9440	. . . . . . . . . . . . . . 15
18	17	ad2antrr 488	. . . . . . . . . . . . . 14 $/\$ $/\$
19		elfzelz 10094	. . . . . . . . . . . . . 14
20		zltp1le 9374	. . . . . . . . . . . . . 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 9445	. . . . . . . . . . . . . 14
26	25	adantr 276	. . . . . . . . . . . . 13 $/\$
27	26	ad2antrr 488	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
28		prmz 12252	. . . . . . . . . . . . . . 15
29		oddm1d2 12036	. . . . . . . . . . . . . . 15
30	28, 29	syl 14	. . . . . . . . . . . . . 14
31	30	biimpa 296	. . . . . . . . . . . . 13 $/\$
32	31	ad2antrr 488	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
33		elfz 10083	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
34	24, 27, 32, 33	syl3anc 1249	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
35		elfzle2 10097	. . . . . . . . . . . . 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 15235	. . . . . . . . . . 11 $/\$
41	39, 19, 40	syl2an 289	. . . . . . . . . 10 $/\$ $/\$ $/\$
42	38, 41	bitrd 188	. . . . . . . . 9 $/\$ $/\$ $/\$
43		2lgslem1a1 15234	. . . . . . . . . . . . 13 $/\$
44	1, 43	sylan 283	. . . . . . . . . . . 12 $/\$
45	44	adantr 276	. . . . . . . . . . 11 $/\$ $/\$
46		oveq1 5926	. . . . . . . . . . . . 13
47	46	oveq1d 5934	. . . . . . . . . . . . 13
48	46, 47	eqeq12d 2208	. . . . . . . . . . . 12
49	48	rspccva 2864	. . . . . . . . . . 11 $/\$
50	45, 49	sylan 283	. . . . . . . . . 10 $/\$ $/\$ $/\$
51	50	breq2d 4042	. . . . . . . . 9 $/\$ $/\$ $/\$
52	42, 51	bitrd 188	. . . . . . . 8 $/\$ $/\$ $/\$
53		oveq1 5926	. . . . . . . . . 10
54	53	eqcomd 2199	. . . . . . . . 9
55	54	breq2d 4042	. . . . . . . 8
56	52, 55	sylan9bb 462	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
57	56	pm5.32da 452	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
58	14, 57	bitrid 192	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
59	58	rexbidva 2491	. . . 4 $/\$ $/\$ $/\$ $/\$
60	13, 59	bitrd 188	. . 3 $/\$ $/\$ $/\$
61	60	bicomd 141	. 2 $/\$ $/\$ $/\$
62	61	rabbidva 2748	1 $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105

wceq 1364

wcel 2164

wral 2472

wrex 2473

crab 2476

wss 3154 class class class wbr 4030

cfv 5255 (class class class)co 5919

c1 7875

caddc 7877

cmul 7879

clt 8056

cle 8057

cmin 8192

cdiv 8693

cn 8984

c2 9035

c4 9037

cn0 9243

cz 9320

cuz 9595

cfz 10077

cfl 10340

cmo 10396

cdvds 11933

cprime 12248

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 ax-pre-mulext 7992 ax-arch 7993

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-po 4328 df-iso 4329 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-div 8694 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-n0 9244 df-z 9321 df-uz 9596 df-q 9688 df-rp 9723 df-fz 10078 df-fl 10342 df-mod 10397 df-dvds 11934 df-prm 12249

This theorem is referenced by: 2lgslem1 15239

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