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Theorem xrmaxiflemcom 10857

Description: Lemma for xrmaxif 10859. Commutativity of an expression which we will later show to be the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.)

**Assertion**
Ref	Expression
xrmaxiflemcom	$/\$

**Proof of Theorem xrmaxiflemcom**
Step	Hyp	Ref	Expression
1		simpr 109	. . . 4 $/\$ $/\$
2	1	iftrued 3428	. . 3 $/\$ $/\$
3		xrpnfdc 9466	. . . . . . 7 DECID
4	3	ad2antrr 475	. . . . . 6 $/\$ $/\$ DECID
5		exmiddc 788	. . . . . 6 DECID $\/$
6	4, 5	syl 14	. . . . 5 $/\$ $/\$ $\/$
7		eqid 2100	. . . . . . . 8
8	7	biantru 298	. . . . . . 7 $/\$
9	8	a1i 9	. . . . . 6 $/\$ $/\$ $/\$
10		xrmnfdc 9467	. . . . . . . . . . 11 DECID
11	10	ad2antrr 475	. . . . . . . . . 10 $/\$ $/\$ DECID
12		exmiddc 788	. . . . . . . . . 10 DECID $\/$
13	11, 12	syl 14	. . . . . . . . 9 $/\$ $/\$ $\/$
14		iba 296	. . . . . . . . . . . 12 $/\$
15	14	eqcoms 2103	. . . . . . . . . . 11 $/\$
16	15	adantl 273	. . . . . . . . . 10 $/\$ $/\$ $/\$
17	1	iftrued 3428	. . . . . . . . . . . 12 $/\$ $/\$
18	17	eqcomd 2105	. . . . . . . . . . 11 $/\$ $/\$
19	18	biantrud 300	. . . . . . . . . 10 $/\$ $/\$ $/\$
20	16, 19	orbi12d 748	. . . . . . . . 9 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
21	13, 20	mpbid 146	. . . . . . . 8 $/\$ $/\$ $/\$ $\/$ $/\$
22		eqifdc 3453	. . . . . . . . 9 DECID $/\$ $\/$ $/\$
23	11, 22	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$ $\/$ $/\$
24	21, 23	mpbird 166	. . . . . . 7 $/\$ $/\$
25	24	biantrud 300	. . . . . 6 $/\$ $/\$ $/\$
26	9, 25	orbi12d 748	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
27	6, 26	mpbid 146	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
28		eqifdc 3453	. . . . 5 DECID $/\$ $\/$ $/\$
29	4, 28	syl 14	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
30	27, 29	mpbird 166	. . 3 $/\$ $/\$
31	2, 30	eqtrd 2132	. 2 $/\$ $/\$
32	3, 5	syl 14	. . . . . . . 8 $\/$
33	32	ad3antrrr 479	. . . . . . 7 $/\$ $/\$ $/\$ $\/$
34		pm4.24 390	. . . . . . . . 9 $/\$
35	34	a1i 9	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
36	10	ad3antrrr 479	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ DECID
37	36, 12	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $\/$
38		simpr 109	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
39	38	eqeq2d 2111	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
40	39	anbi2d 455	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
41		anidm 391	. . . . . . . . . . . . 13 $/\$
42	40, 41	syl6rbb 196	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
43		eqid 2100	. . . . . . . . . . . . . 14
44	43	biantru 298	. . . . . . . . . . . . 13 $/\$
45	44	a1i 9	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
46	42, 45	orbi12d 748	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
47	37, 46	mpbid 146	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
48		eqifdc 3453	. . . . . . . . . . 11 DECID $/\$ $\/$ $/\$
49	36, 48	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
50	47, 49	mpbird 166	. . . . . . . . 9 $/\$ $/\$ $/\$
51	50	biantrud 300	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
52	35, 51	orbi12d 748	. . . . . . 7 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
53	33, 52	mpbid 146	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
54	3	ad3antrrr 479	. . . . . . 7 $/\$ $/\$ $/\$ DECID
55		eqifdc 3453	. . . . . . 7 DECID $/\$ $\/$ $/\$
56	54, 55	syl 14	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
57	53, 56	mpbird 166	. . . . 5 $/\$ $/\$ $/\$
58	38	iftrued 3428	. . . . 5 $/\$ $/\$ $/\$
59	38	iftrued 3428	. . . . . . 7 $/\$ $/\$ $/\$
60	59	ifeq2d 3437	. . . . . 6 $/\$ $/\$ $/\$
61	60	ifeq2d 3437	. . . . 5 $/\$ $/\$ $/\$
62	57, 58, 61	3eqtr4d 2142	. . . 4 $/\$ $/\$ $/\$
63		simpr 109	. . . . . 6 $/\$ $/\$ $/\$
64	63	iffalsed 3431	. . . . 5 $/\$ $/\$ $/\$
65	63	iffalsed 3431	. . . . . . . 8 $/\$ $/\$ $/\$
66	65	ifeq2d 3437	. . . . . . 7 $/\$ $/\$ $/\$
67	66	ifeq2d 3437	. . . . . 6 $/\$ $/\$ $/\$
68		maxcom 10815	. . . . . . 7
69		ifeq2 3425	. . . . . . 7
70		ifeq2 3425	. . . . . . 7
71	68, 69, 70	mp2b 8	. . . . . 6
72	67, 71	syl6eq 2148	. . . . 5 $/\$ $/\$ $/\$
73	64, 72	eqtr4d 2135	. . . 4 $/\$ $/\$ $/\$
74		xrmnfdc 9467	. . . . . 6 DECID
75		exmiddc 788	. . . . . 6 DECID $\/$
76	74, 75	syl 14	. . . . 5 $\/$
77	76	ad2antlr 476	. . . 4 $/\$ $/\$ $\/$
78	62, 73, 77	mpjaodan 753	. . 3 $/\$ $/\$
79		simpr 109	. . . 4 $/\$ $/\$
80	79	iffalsed 3431	. . 3 $/\$ $/\$
81	79	iffalsed 3431	. . . . 5 $/\$ $/\$
82	81	ifeq2d 3437	. . . 4 $/\$ $/\$
83	82	ifeq2d 3437	. . 3 $/\$ $/\$
84	78, 80, 83	3eqtr4d 2142	. 2 $/\$ $/\$
85		xrpnfdc 9466	. . . 4 DECID
86		exmiddc 788	. . . 4 DECID $\/$
87	85, 86	syl 14	. . 3 $\/$
88	87	adantl 273	. 2 $/\$ $\/$
89	31, 84, 88	mpjaodan 753	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 670 DECID wdc 786

wceq 1299

wcel 1448

cif 3421

cpr 3475

csup 6784

cr 7499

cpnf 7669

cmnf 7670

cxr 7671

clt 7672

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587

This theorem depends on definitions: df-bi 116 df-dc 787 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-rab 2384 df-v 2643 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-if 3422 df-pw 3459 df-sn 3480 df-pr 3481 df-uni 3684 df-sup 6786 df-pnf 7674 df-mnf 7675 df-xr 7676

This theorem is referenced by: xrmaxiflemval 10858

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