recexaplem2 - Intuitionistic Logic Explorer

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Theorem recexaplem2 8974

Description: Lemma for recexap 8975. (Contributed by Jim Kingdon, 20-Feb-2020.)

**Assertion**
Ref	Expression
recexaplem2	$/\$ $/\$ # #

**Proof of Theorem recexaplem2**
Step	Hyp	Ref	Expression
1		ax-icn 8268	. . . . . . . . . . 11
2	1	mul01i 8712	. . . . . . . . . 10
3	2	oveq2i 6090	. . . . . . . . 9
4		00id 8461	. . . . . . . . 9
5	3, 4	eqtr2i 2260	. . . . . . . 8
6	5	breq2i 4136	. . . . . . 7 # #
7		0re 8320	. . . . . . . 8
8		apreim 8925	. . . . . . . 8 $/\$ $/\$ $/\$ # # $\/$ #
9	7, 7, 8	mpanr12 443	. . . . . . 7 $/\$ # # $\/$ #
10	6, 9	bitrid 192	. . . . . 6 $/\$ # # $\/$ #
11	10	pm5.32i 458	. . . . 5 $/\$ $/\$ # $/\$ $/\$ # $\/$ #
12		remulcl 8301	. . . . . . . . . 10 $/\$
13	12	anidms 401	. . . . . . . . 9
14		remulcl 8301	. . . . . . . . . 10 $/\$
15	14	anidms 401	. . . . . . . . 9
16	13, 15	anim12i 338	. . . . . . . 8 $/\$ $/\$
17	16	adantr 276	. . . . . . 7 $/\$ $/\$ # $/\$
18		apsqgt0 8923	. . . . . . . . 9 $/\$ #
19		msqge0 8938	. . . . . . . . 9
20	18, 19	anim12i 338	. . . . . . . 8 $/\$ # $/\$ $/\$
21	20	an32s 574	. . . . . . 7 $/\$ $/\$ # $/\$
22		addgtge0 8772	. . . . . . 7 $/\$ $/\$ $/\$
23	17, 21, 22	syl2anc 415	. . . . . 6 $/\$ $/\$ #
24	16	adantr 276	. . . . . . 7 $/\$ $/\$ # $/\$
25		msqge0 8938	. . . . . . . . 9
26		apsqgt0 8923	. . . . . . . . 9 $/\$ #
27	25, 26	anim12i 338	. . . . . . . 8 $/\$ $/\$ # $/\$
28	27	anassrs 404	. . . . . . 7 $/\$ $/\$ # $/\$
29		addgegt0 8771	. . . . . . 7 $/\$ $/\$ $/\$
30	24, 28, 29	syl2anc 415	. . . . . 6 $/\$ $/\$ #
31	23, 30	jaodan 809	. . . . 5 $/\$ $/\$ # $\/$ #
32	11, 31	sylbi 121	. . . 4 $/\$ $/\$ #
33	32	3impa 1225	. . 3 $/\$ $/\$ #
34	33	olcd 746	. 2 $/\$ $/\$ # $\/$
35		simp1 1028	. . . . 5 $/\$ $/\$ #
36	35, 35	remulcld 8350	. . . 4 $/\$ $/\$ #
37		simp2 1029	. . . . 5 $/\$ $/\$ #
38	37, 37	remulcld 8350	. . . 4 $/\$ $/\$ #
39	36, 38	readdcld 8349	. . 3 $/\$ $/\$ #
40		reaplt 8910	. . 3 $/\$ # $\/$
41	39, 7, 40	sylancl 417	. 2 $/\$ $/\$ # # $\/$
42	34, 41	mpbird 167	1 $/\$ $/\$ # #

Colors of variables: wff set class

This proof depends on syntax axioms:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 720 $/\$ w3a 1009

wcel 2209 class class class wbr 4128 (class class class)co 6079

cr 8172

cc0 8173

ci 8175

caddc 8176

cmul 8178

clt 8354

cle 8355 # cap 8903

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4247 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-setind 4682 ax-cnex 8264 ax-resscn 8265 ax-1cn 8266 ax-1re 8267 ax-icn 8268 ax-addcl 8269 ax-addrcl 8270 ax-mulcl 8271 ax-mulrcl 8272 ax-addcom 8273 ax-mulcom 8274 ax-addass 8275 ax-mulass 8276 ax-distr 8277 ax-i2m1 8278 ax-0lt1 8279 ax-1rid 8280 ax-0id 8281 ax-rnegex 8282 ax-precex 8283 ax-cnre 8284 ax-pre-ltirr 8285 ax-pre-ltwlin 8286 ax-pre-lttrn 8287 ax-pre-apti 8288 ax-pre-ltadd 8289 ax-pre-mulgt0 8290 ax-pre-mulext 8291

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3714 df-pr 3715 df-op 3717 df-uni 3934 df-br 4129 df-opab 4191 df-id 4436 df-po 4439 df-iso 4440 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-iota 5335 df-fun 5377 df-fv 5383 df-riota 6032 df-ov 6082 df-oprab 6083 df-mpo 6084 df-pnf 8356 df-mnf 8357 df-xr 8358 df-ltxr 8359 df-le 8360 df-sub 8493 df-neg 8494 df-reap 8897 df-ap 8904

This theorem is used by: recexap 8975

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