dfplpq2 - Intuitionistic Logic Explorer

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Theorem dfplpq2 6338

Description: Alternative definition of pre-addition on positive fractions. (Contributed by Jim Kingdon, 12-Sep-2019.)

**Assertion**
Ref	Expression
dfplpq2	$/\$ $/\$ $/\$ $/\$

Distinct variable group:

**Proof of Theorem dfplpq2**
Step	Hyp	Ref	Expression
1		df-mpt2 5460	. 2 $/\$ $/\$
2		df-plpq 6328	. 2
3		1st2nd2 5743	. . . . . . . . . 10
4	3	eqeq1d 2045	. . . . . . . . 9
5		1st2nd2 5743	. . . . . . . . . 10
6	5	eqeq1d 2045	. . . . . . . . 9
7	4, 6	bi2anan9 538	. . . . . . . 8 $/\$ $/\$ $/\$
8	7	anbi1d 438	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
9		xp1st 5734	. . . . . . . . . . . . . 14
10	9	ad2antlr 458	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
11	7	biimpa 280	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
12	11	simprd 107	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
13		vex 2554	. . . . . . . . . . . . . . . . 17
14		vex 2554	. . . . . . . . . . . . . . . . 17
15	13, 14	opth2 3968	. . . . . . . . . . . . . . . 16 $/\$
16	15	simplbi 259	. . . . . . . . . . . . . . 15
17	16	eleq1d 2103	. . . . . . . . . . . . . 14
18	12, 17	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
19	10, 18	mpbid 135	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
20		xp2nd 5735	. . . . . . . . . . . . . 14
21	20	ad2antrr 457	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
22	11	simpld 105	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
23		vex 2554	. . . . . . . . . . . . . . . . 17
24		vex 2554	. . . . . . . . . . . . . . . . 17
25	23, 24	opth2 3968	. . . . . . . . . . . . . . . 16 $/\$
26	25	simprbi 260	. . . . . . . . . . . . . . 15
27	26	eleq1d 2103	. . . . . . . . . . . . . 14
28	22, 27	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
29	21, 28	mpbid 135	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
30		mulcompig 6315	. . . . . . . . . . . 12 $/\$
31	19, 29, 30	syl2anc 391	. . . . . . . . . . 11 $/\$ $/\$ $/\$
32	31	oveq2d 5471	. . . . . . . . . 10 $/\$ $/\$ $/\$
33	32	opeq1d 3546	. . . . . . . . 9 $/\$ $/\$ $/\$
34	33	eqeq2d 2048	. . . . . . . 8 $/\$ $/\$ $/\$
35	34	pm5.32da 425	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
36	8, 35	bitr3d 179	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
37	36	4exbidv 1747	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
38		xp1st 5734	. . . . . . 7
39	38, 20	jca 290	. . . . . 6 $/\$
40		xp2nd 5735	. . . . . . 7
41	9, 40	jca 290	. . . . . 6 $/\$
42		simpll 481	. . . . . . . . . . 11 $/\$ $/\$ $/\$
43		simprr 484	. . . . . . . . . . 11 $/\$ $/\$ $/\$
44	42, 43	oveq12d 5473	. . . . . . . . . 10 $/\$ $/\$ $/\$
45		simprl 483	. . . . . . . . . . 11 $/\$ $/\$ $/\$
46		simplr 482	. . . . . . . . . . 11 $/\$ $/\$ $/\$
47	45, 46	oveq12d 5473	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	44, 47	oveq12d 5473	. . . . . . . . 9 $/\$ $/\$ $/\$
49	46, 43	oveq12d 5473	. . . . . . . . 9 $/\$ $/\$ $/\$
50	48, 49	opeq12d 3548	. . . . . . . 8 $/\$ $/\$ $/\$
51	50	eqeq2d 2048	. . . . . . 7 $/\$ $/\$ $/\$
52	51	copsex4g 3975	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
53	39, 41, 52	syl2an 273	. . . . 5 $/\$ $/\$ $/\$
54	37, 53	bitr3d 179	. . . 4 $/\$ $/\$ $/\$
55	54	pm5.32i 427	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	55	oprabbii 5502	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57	1, 2, 56	3eqtr4i 2067	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints: $/\$ wa 97

wb 98

wceq 1242

wex 1378

wcel 1390

cop 3370

cxp 4286

cfv 4845 (class class class)co 5455

coprab 5456

cmpt2 5457

c1st 5707

c2nd 5708

cnpi 6256

cpli 6257

cmi 6258

cplpq 6260

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bnd 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254

This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-id 4021 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-oadd 5944 df-omul 5945 df-ni 6288 df-mi 6290 df-plpq 6328

This theorem is referenced by: addpipqqs 6354

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