2lgslem1a - Intuitionistic Logic Explorer

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Theorem 2lgslem1a 15820

Description: Lemma 1 for 2lgslem1 15823. (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 12684	. . . . . . . . . 10
2	1	nnnn0d 9455	. . . . . . . . 9
3	2	ad2antrr 488	. . . . . . . 8 $/\$ $/\$
4		4nn 9307	. . . . . . . 8
5	3, 4	jctir 313	. . . . . . 7 $/\$ $/\$ $/\$
6		fldivnn0 10556	. . . . . . 7 $/\$
7		nn0p1nn 9441	. . . . . . 7
8	5, 6, 7	3syl 17	. . . . . 6 $/\$ $/\$
9		elnnuz 9793	. . . . . 6
10	8, 9	sylib 122	. . . . 5 $/\$ $/\$
11		fzss1 10298	. . . . 5
12		rexss 3294	. . . . 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 9600	. . . . . . . . . . . . . . 15
18	17	ad2antrr 488	. . . . . . . . . . . . . 14 $/\$ $/\$
19		elfzelz 10260	. . . . . . . . . . . . . 14
20		zltp1le 9534	. . . . . . . . . . . . . 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 9605	. . . . . . . . . . . . . 14
26	25	adantr 276	. . . . . . . . . . . . 13 $/\$
27	26	ad2antrr 488	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
28		prmz 12685	. . . . . . . . . . . . . . 15
29		oddm1d2 12455	. . . . . . . . . . . . . . 15
30	28, 29	syl 14	. . . . . . . . . . . . . 14
31	30	biimpa 296	. . . . . . . . . . . . 13 $/\$
32	31	ad2antrr 488	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
33		elfz 10249	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
34	24, 27, 32, 33	syl3anc 1273	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
35		elfzle2 10263	. . . . . . . . . . . . 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 15819	. . . . . . . . . . 11 $/\$
41	39, 19, 40	syl2an 289	. . . . . . . . . 10 $/\$ $/\$ $/\$
42	38, 41	bitrd 188	. . . . . . . . 9 $/\$ $/\$ $/\$
43		2lgslem1a1 15818	. . . . . . . . . . . . 13 $/\$
44	1, 43	sylan 283	. . . . . . . . . . . 12 $/\$
45	44	adantr 276	. . . . . . . . . . 11 $/\$ $/\$
46		oveq1 6025	. . . . . . . . . . . . 13
47	46	oveq1d 6033	. . . . . . . . . . . . 13
48	46, 47	eqeq12d 2246	. . . . . . . . . . . 12
49	48	rspccva 2909	. . . . . . . . . . 11 $/\$
50	45, 49	sylan 283	. . . . . . . . . 10 $/\$ $/\$ $/\$
51	50	breq2d 4100	. . . . . . . . 9 $/\$ $/\$ $/\$
52	42, 51	bitrd 188	. . . . . . . 8 $/\$ $/\$ $/\$
53		oveq1 6025	. . . . . . . . . 10
54	53	eqcomd 2237	. . . . . . . . 9
55	54	breq2d 4100	. . . . . . . 8
56	52, 55	sylan9bb 462	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
57	56	pm5.32da 452	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
58	14, 57	bitrid 192	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
59	58	rexbidva 2529	. . . 4 $/\$ $/\$ $/\$ $/\$
60	13, 59	bitrd 188	. . 3 $/\$ $/\$ $/\$
61	60	bicomd 141	. 2 $/\$ $/\$ $/\$
62	61	rabbidva 2790	1 $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105

wceq 1397

wcel 2202

wral 2510

wrex 2511

crab 2514

wss 3200 class class class wbr 4088

cfv 5326 (class class class)co 6018

c1 8033

caddc 8035

cmul 8037

clt 8214

cle 8215

cmin 8350

cdiv 8852

cn 9143

c2 9194

c4 9196

cn0 9402

cz 9479

cuz 9755

cfz 10243

cfl 10529

cmo 10585

cdvds 12350

cprime 12681

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-pre-mulext 8150 ax-arch 8151

This theorem depends on definitions: df-bi 117 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-xor 1420 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-id 4390 df-po 4393 df-iso 4394 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-fv 5334 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-1st 6303 df-2nd 6304 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-div 8853 df-inn 9144 df-2 9202 df-3 9203 df-4 9204 df-n0 9403 df-z 9480 df-uz 9756 df-q 9854 df-rp 9889 df-fz 10244 df-fl 10531 df-mod 10586 df-dvds 12351 df-prm 12682

This theorem is referenced by: 2lgslem1 15823

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