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Theorem xrmaxiflemcom 11212

Description: Lemma for xrmaxif 11214. 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 3533	. . 3 $/\$ $/\$
3		xrpnfdc 9799	. . . . . . 7 DECID
4	3	ad2antrr 485	. . . . . 6 $/\$ $/\$ DECID
5		exmiddc 831	. . . . . 6 DECID $\/$
6	4, 5	syl 14	. . . . 5 $/\$ $/\$ $\/$
7		eqid 2170	. . . . . . . 8
8	7	biantru 300	. . . . . . 7 $/\$
9	8	a1i 9	. . . . . 6 $/\$ $/\$ $/\$
10		xrmnfdc 9800	. . . . . . . . . . 11 DECID
11	10	ad2antrr 485	. . . . . . . . . 10 $/\$ $/\$ DECID
12		exmiddc 831	. . . . . . . . . 10 DECID $\/$
13	11, 12	syl 14	. . . . . . . . 9 $/\$ $/\$ $\/$
14		iba 298	. . . . . . . . . . . 12 $/\$
15	14	eqcoms 2173	. . . . . . . . . . 11 $/\$
16	15	adantl 275	. . . . . . . . . 10 $/\$ $/\$ $/\$
17	1	iftrued 3533	. . . . . . . . . . . 12 $/\$ $/\$
18	17	eqcomd 2176	. . . . . . . . . . 11 $/\$ $/\$
19	18	biantrud 302	. . . . . . . . . 10 $/\$ $/\$ $/\$
20	16, 19	orbi12d 788	. . . . . . . . 9 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
21	13, 20	mpbid 146	. . . . . . . 8 $/\$ $/\$ $/\$ $\/$ $/\$
22		eqifdc 3560	. . . . . . . . 9 DECID $/\$ $\/$ $/\$
23	11, 22	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$ $\/$ $/\$
24	21, 23	mpbird 166	. . . . . . 7 $/\$ $/\$
25	24	biantrud 302	. . . . . 6 $/\$ $/\$ $/\$
26	9, 25	orbi12d 788	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
27	6, 26	mpbid 146	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
28		eqifdc 3560	. . . . 5 DECID $/\$ $\/$ $/\$
29	4, 28	syl 14	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
30	27, 29	mpbird 166	. . 3 $/\$ $/\$
31	2, 30	eqtrd 2203	. 2 $/\$ $/\$
32	3, 5	syl 14	. . . . . . . 8 $\/$
33	32	ad3antrrr 489	. . . . . . 7 $/\$ $/\$ $/\$ $\/$
34		pm4.24 393	. . . . . . . . 9 $/\$
35	34	a1i 9	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
36	10	ad3antrrr 489	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ DECID
37	36, 12	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $\/$
38		simpr 109	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
39	38	eqeq2d 2182	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
40	39	anbi2d 461	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
41		anidm 394	. . . . . . . . . . . . 13 $/\$
42	40, 41	bitr2di 196	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
43		eqid 2170	. . . . . . . . . . . . . 14
44	43	biantru 300	. . . . . . . . . . . . 13 $/\$
45	44	a1i 9	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
46	42, 45	orbi12d 788	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
47	37, 46	mpbid 146	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
48		eqifdc 3560	. . . . . . . . . . 11 DECID $/\$ $\/$ $/\$
49	36, 48	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
50	47, 49	mpbird 166	. . . . . . . . 9 $/\$ $/\$ $/\$
51	50	biantrud 302	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
52	35, 51	orbi12d 788	. . . . . . 7 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
53	33, 52	mpbid 146	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
54	3	ad3antrrr 489	. . . . . . 7 $/\$ $/\$ $/\$ DECID
55		eqifdc 3560	. . . . . . 7 DECID $/\$ $\/$ $/\$
56	54, 55	syl 14	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
57	53, 56	mpbird 166	. . . . 5 $/\$ $/\$ $/\$
58	38	iftrued 3533	. . . . 5 $/\$ $/\$ $/\$
59	38	iftrued 3533	. . . . . . 7 $/\$ $/\$ $/\$
60	59	ifeq2d 3544	. . . . . 6 $/\$ $/\$ $/\$
61	60	ifeq2d 3544	. . . . 5 $/\$ $/\$ $/\$
62	57, 58, 61	3eqtr4d 2213	. . . 4 $/\$ $/\$ $/\$
63		simpr 109	. . . . . 6 $/\$ $/\$ $/\$
64	63	iffalsed 3536	. . . . 5 $/\$ $/\$ $/\$
65	63	iffalsed 3536	. . . . . . . 8 $/\$ $/\$ $/\$
66	65	ifeq2d 3544	. . . . . . 7 $/\$ $/\$ $/\$
67	66	ifeq2d 3544	. . . . . 6 $/\$ $/\$ $/\$
68		maxcom 11167	. . . . . . 7
69		ifeq2 3530	. . . . . . 7
70		ifeq2 3530	. . . . . . 7
71	68, 69, 70	mp2b 8	. . . . . 6
72	67, 71	eqtrdi 2219	. . . . 5 $/\$ $/\$ $/\$
73	64, 72	eqtr4d 2206	. . . 4 $/\$ $/\$ $/\$
74		xrmnfdc 9800	. . . . . 6 DECID
75		exmiddc 831	. . . . . 6 DECID $\/$
76	74, 75	syl 14	. . . . 5 $\/$
77	76	ad2antlr 486	. . . 4 $/\$ $/\$ $\/$
78	62, 73, 77	mpjaodan 793	. . 3 $/\$ $/\$
79		simpr 109	. . . 4 $/\$ $/\$
80	79	iffalsed 3536	. . 3 $/\$ $/\$
81	79	iffalsed 3536	. . . . 5 $/\$ $/\$
82	81	ifeq2d 3544	. . . 4 $/\$ $/\$
83	82	ifeq2d 3544	. . 3 $/\$ $/\$
84	78, 80, 83	3eqtr4d 2213	. 2 $/\$ $/\$
85		xrpnfdc 9799	. . . 4 DECID
86		exmiddc 831	. . . 4 DECID $\/$
87	85, 86	syl 14	. . 3 $\/$
88	87	adantl 275	. 2 $/\$ $\/$
89	31, 84, 88	mpjaodan 793	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 703 DECID wdc 829

wceq 1348

wcel 2141

cif 3526

cpr 3584

csup 6959

cr 7773

cpnf 7951

cmnf 7952

cxr 7953

clt 7954

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-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 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-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-uni 3797 df-sup 6961 df-pnf 7956 df-mnf 7957 df-xr 7958

This theorem is referenced by: xrmaxiflemval 11213

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