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Theorem xrmaxiflemcom 11011

Description: Lemma for xrmaxif 11013. 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 3476	. . 3 $/\$ $/\$
3		xrpnfdc 9618	. . . . . . 7 DECID
4	3	ad2antrr 479	. . . . . 6 $/\$ $/\$ DECID
5		exmiddc 821	. . . . . 6 DECID $\/$
6	4, 5	syl 14	. . . . 5 $/\$ $/\$ $\/$
7		eqid 2137	. . . . . . . 8
8	7	biantru 300	. . . . . . 7 $/\$
9	8	a1i 9	. . . . . 6 $/\$ $/\$ $/\$
10		xrmnfdc 9619	. . . . . . . . . . 11 DECID
11	10	ad2antrr 479	. . . . . . . . . 10 $/\$ $/\$ DECID
12		exmiddc 821	. . . . . . . . . 10 DECID $\/$
13	11, 12	syl 14	. . . . . . . . 9 $/\$ $/\$ $\/$
14		iba 298	. . . . . . . . . . . 12 $/\$
15	14	eqcoms 2140	. . . . . . . . . . 11 $/\$
16	15	adantl 275	. . . . . . . . . 10 $/\$ $/\$ $/\$
17	1	iftrued 3476	. . . . . . . . . . . 12 $/\$ $/\$
18	17	eqcomd 2143	. . . . . . . . . . 11 $/\$ $/\$
19	18	biantrud 302	. . . . . . . . . 10 $/\$ $/\$ $/\$
20	16, 19	orbi12d 782	. . . . . . . . 9 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
21	13, 20	mpbid 146	. . . . . . . 8 $/\$ $/\$ $/\$ $\/$ $/\$
22		eqifdc 3501	. . . . . . . . 9 DECID $/\$ $\/$ $/\$
23	11, 22	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$ $\/$ $/\$
24	21, 23	mpbird 166	. . . . . . 7 $/\$ $/\$
25	24	biantrud 302	. . . . . 6 $/\$ $/\$ $/\$
26	9, 25	orbi12d 782	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
27	6, 26	mpbid 146	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
28		eqifdc 3501	. . . . 5 DECID $/\$ $\/$ $/\$
29	4, 28	syl 14	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
30	27, 29	mpbird 166	. . 3 $/\$ $/\$
31	2, 30	eqtrd 2170	. 2 $/\$ $/\$
32	3, 5	syl 14	. . . . . . . 8 $\/$
33	32	ad3antrrr 483	. . . . . . 7 $/\$ $/\$ $/\$ $\/$
34		pm4.24 392	. . . . . . . . 9 $/\$
35	34	a1i 9	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
36	10	ad3antrrr 483	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ DECID
37	36, 12	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $\/$
38		simpr 109	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
39	38	eqeq2d 2149	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
40	39	anbi2d 459	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
41		anidm 393	. . . . . . . . . . . . 13 $/\$
42	40, 41	syl6rbb 196	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
43		eqid 2137	. . . . . . . . . . . . . 14
44	43	biantru 300	. . . . . . . . . . . . 13 $/\$
45	44	a1i 9	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
46	42, 45	orbi12d 782	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
47	37, 46	mpbid 146	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
48		eqifdc 3501	. . . . . . . . . . 11 DECID $/\$ $\/$ $/\$
49	36, 48	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
50	47, 49	mpbird 166	. . . . . . . . 9 $/\$ $/\$ $/\$
51	50	biantrud 302	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
52	35, 51	orbi12d 782	. . . . . . 7 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
53	33, 52	mpbid 146	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
54	3	ad3antrrr 483	. . . . . . 7 $/\$ $/\$ $/\$ DECID
55		eqifdc 3501	. . . . . . 7 DECID $/\$ $\/$ $/\$
56	54, 55	syl 14	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
57	53, 56	mpbird 166	. . . . 5 $/\$ $/\$ $/\$
58	38	iftrued 3476	. . . . 5 $/\$ $/\$ $/\$
59	38	iftrued 3476	. . . . . . 7 $/\$ $/\$ $/\$
60	59	ifeq2d 3485	. . . . . 6 $/\$ $/\$ $/\$
61	60	ifeq2d 3485	. . . . 5 $/\$ $/\$ $/\$
62	57, 58, 61	3eqtr4d 2180	. . . 4 $/\$ $/\$ $/\$
63		simpr 109	. . . . . 6 $/\$ $/\$ $/\$
64	63	iffalsed 3479	. . . . 5 $/\$ $/\$ $/\$
65	63	iffalsed 3479	. . . . . . . 8 $/\$ $/\$ $/\$
66	65	ifeq2d 3485	. . . . . . 7 $/\$ $/\$ $/\$
67	66	ifeq2d 3485	. . . . . 6 $/\$ $/\$ $/\$
68		maxcom 10968	. . . . . . 7
69		ifeq2 3473	. . . . . . 7
70		ifeq2 3473	. . . . . . 7
71	68, 69, 70	mp2b 8	. . . . . 6
72	67, 71	syl6eq 2186	. . . . 5 $/\$ $/\$ $/\$
73	64, 72	eqtr4d 2173	. . . 4 $/\$ $/\$ $/\$
74		xrmnfdc 9619	. . . . . 6 DECID
75		exmiddc 821	. . . . . 6 DECID $\/$
76	74, 75	syl 14	. . . . 5 $\/$
77	76	ad2antlr 480	. . . 4 $/\$ $/\$ $\/$
78	62, 73, 77	mpjaodan 787	. . 3 $/\$ $/\$
79		simpr 109	. . . 4 $/\$ $/\$
80	79	iffalsed 3479	. . 3 $/\$ $/\$
81	79	iffalsed 3479	. . . . 5 $/\$ $/\$
82	81	ifeq2d 3485	. . . 4 $/\$ $/\$
83	82	ifeq2d 3485	. . 3 $/\$ $/\$
84	78, 80, 83	3eqtr4d 2180	. 2 $/\$ $/\$
85		xrpnfdc 9618	. . . 4 DECID
86		exmiddc 821	. . . 4 DECID $\/$
87	85, 86	syl 14	. . 3 $\/$
88	87	adantl 275	. 2 $/\$ $\/$
89	31, 84, 88	mpjaodan 787	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 697 DECID wdc 819

wceq 1331

wcel 1480

cif 3469

cpr 3523

csup 6862

cr 7612

cpnf 7790

cmnf 7791

cxr 7792

clt 7793

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-uni 3732 df-sup 6864 df-pnf 7795 df-mnf 7796 df-xr 7797

This theorem is referenced by: xrmaxiflemval 11012

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