recexaplem2 - Intuitionistic Logic Explorer

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Theorem recexaplem2 8570

Description: Lemma for recexap 8571. (Contributed by Jim Kingdon, 20-Feb-2020.)

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

**Proof of Theorem recexaplem2**
Step	Hyp	Ref	Expression
1		ax-icn 7869	. . . . . . . . . . 11
2	1	mul01i 8310	. . . . . . . . . 10
3	2	oveq2i 5864	. . . . . . . . 9
4		00id 8060	. . . . . . . . 9
5	3, 4	eqtr2i 2192	. . . . . . . 8
6	5	breq2i 3997	. . . . . . 7 # #
7		0re 7920	. . . . . . . 8
8		apreim 8522	. . . . . . . 8 $/\$ $/\$ $/\$ # # $\/$ #
9	7, 7, 8	mpanr12 437	. . . . . . 7 $/\$ # # $\/$ #
10	6, 9	syl5bb 191	. . . . . 6 $/\$ # # $\/$ #
11	10	pm5.32i 451	. . . . 5 $/\$ $/\$ # $/\$ $/\$ # $\/$ #
12		remulcl 7902	. . . . . . . . . 10 $/\$
13	12	anidms 395	. . . . . . . . 9
14		remulcl 7902	. . . . . . . . . 10 $/\$
15	14	anidms 395	. . . . . . . . 9
16	13, 15	anim12i 336	. . . . . . . 8 $/\$ $/\$
17	16	adantr 274	. . . . . . 7 $/\$ $/\$ # $/\$
18		apsqgt0 8520	. . . . . . . . 9 $/\$ #
19		msqge0 8535	. . . . . . . . 9
20	18, 19	anim12i 336	. . . . . . . 8 $/\$ # $/\$ $/\$
21	20	an32s 563	. . . . . . 7 $/\$ $/\$ # $/\$
22		addgtge0 8369	. . . . . . 7 $/\$ $/\$ $/\$
23	17, 21, 22	syl2anc 409	. . . . . 6 $/\$ $/\$ #
24	16	adantr 274	. . . . . . 7 $/\$ $/\$ # $/\$
25		msqge0 8535	. . . . . . . . 9
26		apsqgt0 8520	. . . . . . . . 9 $/\$ #
27	25, 26	anim12i 336	. . . . . . . 8 $/\$ $/\$ # $/\$
28	27	anassrs 398	. . . . . . 7 $/\$ $/\$ # $/\$
29		addgegt0 8368	. . . . . . 7 $/\$ $/\$ $/\$
30	24, 28, 29	syl2anc 409	. . . . . 6 $/\$ $/\$ #
31	23, 30	jaodan 792	. . . . 5 $/\$ $/\$ # $\/$ #
32	11, 31	sylbi 120	. . . 4 $/\$ $/\$ #
33	32	3impa 1189	. . 3 $/\$ $/\$ #
34	33	olcd 729	. 2 $/\$ $/\$ # $\/$
35		simp1 992	. . . . 5 $/\$ $/\$ #
36	35, 35	remulcld 7950	. . . 4 $/\$ $/\$ #
37		simp2 993	. . . . 5 $/\$ $/\$ #
38	37, 37	remulcld 7950	. . . 4 $/\$ $/\$ #
39	36, 38	readdcld 7949	. . 3 $/\$ $/\$ #
40		reaplt 8507	. . 3 $/\$ # $\/$
41	39, 7, 40	sylancl 411	. 2 $/\$ $/\$ # # $\/$
42	34, 41	mpbird 166	1 $/\$ $/\$ # #

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $\/$ wo 703 $/\$ w3a 973

wcel 2141 class class class wbr 3989 (class class class)co 5853

cr 7773

cc0 7774

ci 7776

caddc 7777

cmul 7779

clt 7954

cle 7955 # cap 8500

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-po 4281 df-iso 4282 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501

This theorem is referenced by: recexap 8571

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