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Theorem xrmaxiflemcom 11560

Description: Lemma for xrmaxif 11562. 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 110	. . . 4 $/\$ $/\$
2	1	iftrued 3578	. . 3 $/\$ $/\$
3		xrpnfdc 9964	. . . . . . 7 DECID
4	3	ad2antrr 488	. . . . . 6 $/\$ $/\$ DECID
5		exmiddc 838	. . . . . 6 DECID $\/$
6	4, 5	syl 14	. . . . 5 $/\$ $/\$ $\/$
7		eqid 2205	. . . . . . . 8
8	7	biantru 302	. . . . . . 7 $/\$
9	8	a1i 9	. . . . . 6 $/\$ $/\$ $/\$
10		xrmnfdc 9965	. . . . . . . . . . 11 DECID
11	10	ad2antrr 488	. . . . . . . . . 10 $/\$ $/\$ DECID
12		exmiddc 838	. . . . . . . . . 10 DECID $\/$
13	11, 12	syl 14	. . . . . . . . 9 $/\$ $/\$ $\/$
14		iba 300	. . . . . . . . . . . 12 $/\$
15	14	eqcoms 2208	. . . . . . . . . . 11 $/\$
16	15	adantl 277	. . . . . . . . . 10 $/\$ $/\$ $/\$
17	1	iftrued 3578	. . . . . . . . . . . 12 $/\$ $/\$
18	17	eqcomd 2211	. . . . . . . . . . 11 $/\$ $/\$
19	18	biantrud 304	. . . . . . . . . 10 $/\$ $/\$ $/\$
20	16, 19	orbi12d 795	. . . . . . . . 9 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
21	13, 20	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$ $\/$ $/\$
22		eqifdc 3607	. . . . . . . . 9 DECID $/\$ $\/$ $/\$
23	11, 22	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$ $\/$ $/\$
24	21, 23	mpbird 167	. . . . . . 7 $/\$ $/\$
25	24	biantrud 304	. . . . . 6 $/\$ $/\$ $/\$
26	9, 25	orbi12d 795	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
27	6, 26	mpbid 147	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
28		eqifdc 3607	. . . . 5 DECID $/\$ $\/$ $/\$
29	4, 28	syl 14	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
30	27, 29	mpbird 167	. . 3 $/\$ $/\$
31	2, 30	eqtrd 2238	. 2 $/\$ $/\$
32	3, 5	syl 14	. . . . . . . 8 $\/$
33	32	ad3antrrr 492	. . . . . . 7 $/\$ $/\$ $/\$ $\/$
34		pm4.24 395	. . . . . . . . 9 $/\$
35	34	a1i 9	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
36	10	ad3antrrr 492	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ DECID
37	36, 12	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $\/$
38		simpr 110	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
39	38	eqeq2d 2217	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
40	39	anbi2d 464	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
41		anidm 396	. . . . . . . . . . . . 13 $/\$
42	40, 41	bitr2di 197	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
43		eqid 2205	. . . . . . . . . . . . . 14
44	43	biantru 302	. . . . . . . . . . . . 13 $/\$
45	44	a1i 9	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
46	42, 45	orbi12d 795	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
47	37, 46	mpbid 147	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
48		eqifdc 3607	. . . . . . . . . . 11 DECID $/\$ $\/$ $/\$
49	36, 48	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
50	47, 49	mpbird 167	. . . . . . . . 9 $/\$ $/\$ $/\$
51	50	biantrud 304	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
52	35, 51	orbi12d 795	. . . . . . 7 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
53	33, 52	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
54	3	ad3antrrr 492	. . . . . . 7 $/\$ $/\$ $/\$ DECID
55		eqifdc 3607	. . . . . . 7 DECID $/\$ $\/$ $/\$
56	54, 55	syl 14	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
57	53, 56	mpbird 167	. . . . 5 $/\$ $/\$ $/\$
58	38	iftrued 3578	. . . . 5 $/\$ $/\$ $/\$
59	38	iftrued 3578	. . . . . . 7 $/\$ $/\$ $/\$
60	59	ifeq2d 3589	. . . . . 6 $/\$ $/\$ $/\$
61	60	ifeq2d 3589	. . . . 5 $/\$ $/\$ $/\$
62	57, 58, 61	3eqtr4d 2248	. . . 4 $/\$ $/\$ $/\$
63		simpr 110	. . . . . 6 $/\$ $/\$ $/\$
64	63	iffalsed 3581	. . . . 5 $/\$ $/\$ $/\$
65	63	iffalsed 3581	. . . . . . . 8 $/\$ $/\$ $/\$
66	65	ifeq2d 3589	. . . . . . 7 $/\$ $/\$ $/\$
67	66	ifeq2d 3589	. . . . . 6 $/\$ $/\$ $/\$
68		maxcom 11514	. . . . . . 7
69		ifeq2 3575	. . . . . . 7
70		ifeq2 3575	. . . . . . 7
71	68, 69, 70	mp2b 8	. . . . . 6
72	67, 71	eqtrdi 2254	. . . . 5 $/\$ $/\$ $/\$
73	64, 72	eqtr4d 2241	. . . 4 $/\$ $/\$ $/\$
74		xrmnfdc 9965	. . . . . 6 DECID
75		exmiddc 838	. . . . . 6 DECID $\/$
76	74, 75	syl 14	. . . . 5 $\/$
77	76	ad2antlr 489	. . . 4 $/\$ $/\$ $\/$
78	62, 73, 77	mpjaodan 800	. . 3 $/\$ $/\$
79		simpr 110	. . . 4 $/\$ $/\$
80	79	iffalsed 3581	. . 3 $/\$ $/\$
81	79	iffalsed 3581	. . . . 5 $/\$ $/\$
82	81	ifeq2d 3589	. . . 4 $/\$ $/\$
83	82	ifeq2d 3589	. . 3 $/\$ $/\$
84	78, 80, 83	3eqtr4d 2248	. 2 $/\$ $/\$
85		xrpnfdc 9964	. . . 4 DECID
86		exmiddc 838	. . . 4 DECID $\/$
87	85, 86	syl 14	. . 3 $\/$
88	87	adantl 277	. 2 $/\$ $\/$
89	31, 84, 88	mpjaodan 800	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 710 DECID wdc 836

wceq 1373

wcel 2176

cif 3571

cpr 3634

csup 7084

cr 7924

cpnf 8104

cmnf 8105

cxr 8106

clt 8107

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-un 4480 ax-setind 4585 ax-cnex 8016 ax-resscn 8017

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-rab 2493 df-v 2774 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-uni 3851 df-sup 7086 df-pnf 8109 df-mnf 8110 df-xr 8111

This theorem is referenced by: xrmaxiflemval 11561

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