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Theorem xrmaxiflemcom 11993

Description: Lemma for xrmaxif 11995. 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 3644	. . 3 $/\$ $/\$
3		xrpnfdc 10223	. . . . . . 7 DECID
4	3	ad2antrr 492	. . . . . 6 $/\$ $/\$ DECID
5		exmiddc 848	. . . . . 6 DECID $\/$
6	4, 5	syl 14	. . . . 5 $/\$ $/\$ $\/$
7		eqid 2238	. . . . . . . 8
8	7	biantru 302	. . . . . . 7 $/\$
9	8	a1i 9	. . . . . 6 $/\$ $/\$ $/\$
10		xrmnfdc 10224	. . . . . . . . . . 11 DECID
11	10	ad2antrr 492	. . . . . . . . . 10 $/\$ $/\$ DECID
12		exmiddc 848	. . . . . . . . . 10 DECID $\/$
13	11, 12	syl 14	. . . . . . . . 9 $/\$ $/\$ $\/$
14		iba 300	. . . . . . . . . . . 12 $/\$
15	14	eqcoms 2241	. . . . . . . . . . 11 $/\$
16	15	adantl 277	. . . . . . . . . 10 $/\$ $/\$ $/\$
17	1	iftrued 3644	. . . . . . . . . . . 12 $/\$ $/\$
18	17	eqcomd 2244	. . . . . . . . . . 11 $/\$ $/\$
19	18	biantrud 304	. . . . . . . . . 10 $/\$ $/\$ $/\$
20	16, 19	orbi12d 805	. . . . . . . . 9 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
21	13, 20	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$ $\/$ $/\$
22		eqifdc 3674	. . . . . . . . 9 DECID $/\$ $\/$ $/\$
23	11, 22	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$ $\/$ $/\$
24	21, 23	mpbird 167	. . . . . . 7 $/\$ $/\$
25	24	biantrud 304	. . . . . 6 $/\$ $/\$ $/\$
26	9, 25	orbi12d 805	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
27	6, 26	mpbid 147	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
28		eqifdc 3674	. . . . 5 DECID $/\$ $\/$ $/\$
29	4, 28	syl 14	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
30	27, 29	mpbird 167	. . 3 $/\$ $/\$
31	2, 30	eqtrd 2271	. 2 $/\$ $/\$
32	3, 5	syl 14	. . . . . . . 8 $\/$
33	32	ad3antrrr 496	. . . . . . 7 $/\$ $/\$ $/\$ $\/$
34		pm4.24 399	. . . . . . . . 9 $/\$
35	34	a1i 9	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
36	10	ad3antrrr 496	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ DECID
37	36, 12	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $\/$
38		simpr 110	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
39	38	eqeq2d 2250	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
40	39	anbi2d 468	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
41		anidm 400	. . . . . . . . . . . . 13 $/\$
42	40, 41	bitr2di 197	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
43		eqid 2238	. . . . . . . . . . . . . 14
44	43	biantru 302	. . . . . . . . . . . . 13 $/\$
45	44	a1i 9	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
46	42, 45	orbi12d 805	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
47	37, 46	mpbid 147	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
48		eqifdc 3674	. . . . . . . . . . 11 DECID $/\$ $\/$ $/\$
49	36, 48	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
50	47, 49	mpbird 167	. . . . . . . . 9 $/\$ $/\$ $/\$
51	50	biantrud 304	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
52	35, 51	orbi12d 805	. . . . . . 7 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
53	33, 52	mpbid 147	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
54	3	ad3antrrr 496	. . . . . . 7 $/\$ $/\$ $/\$ DECID
55		eqifdc 3674	. . . . . . 7 DECID $/\$ $\/$ $/\$
56	54, 55	syl 14	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
57	53, 56	mpbird 167	. . . . 5 $/\$ $/\$ $/\$
58	38	iftrued 3644	. . . . 5 $/\$ $/\$ $/\$
59	38	iftrued 3644	. . . . . . 7 $/\$ $/\$ $/\$
60	59	ifeq2d 3656	. . . . . 6 $/\$ $/\$ $/\$
61	60	ifeq2d 3656	. . . . 5 $/\$ $/\$ $/\$
62	57, 58, 61	3eqtr4d 2281	. . . 4 $/\$ $/\$ $/\$
63		simpr 110	. . . . . 6 $/\$ $/\$ $/\$
64	63	iffalsed 3647	. . . . 5 $/\$ $/\$ $/\$
65	63	iffalsed 3647	. . . . . . . 8 $/\$ $/\$ $/\$
66	65	ifeq2d 3656	. . . . . . 7 $/\$ $/\$ $/\$
67	66	ifeq2d 3656	. . . . . 6 $/\$ $/\$ $/\$
68		maxcom 11947	. . . . . . 7
69		ifeq2 3641	. . . . . . 7
70		ifeq2 3641	. . . . . . 7
71	68, 69, 70	mp2b 8	. . . . . 6
72	67, 71	eqtrdi 2287	. . . . 5 $/\$ $/\$ $/\$
73	64, 72	eqtr4d 2274	. . . 4 $/\$ $/\$ $/\$
74		xrmnfdc 10224	. . . . . 6 DECID
75		exmiddc 848	. . . . . 6 DECID $\/$
76	74, 75	syl 14	. . . . 5 $\/$
77	76	ad2antlr 493	. . . 4 $/\$ $/\$ $\/$
78	62, 73, 77	mpjaodan 810	. . 3 $/\$ $/\$
79		simpr 110	. . . 4 $/\$ $/\$
80	79	iffalsed 3647	. . 3 $/\$ $/\$
81	79	iffalsed 3647	. . . . 5 $/\$ $/\$
82	81	ifeq2d 3656	. . . 4 $/\$ $/\$
83	82	ifeq2d 3656	. . 3 $/\$ $/\$
84	78, 80, 83	3eqtr4d 2281	. 2 $/\$ $/\$
85		xrpnfdc 10223	. . . 4 DECID
86		exmiddc 848	. . . 4 DECID $\/$
87	85, 86	syl 14	. . 3 $\/$
88	87	adantl 277	. 2 $/\$ $\/$
89	31, 84, 88	mpjaodan 810	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 720 DECID wdc 846

wceq 1402

wcel 2209

cif 3635

cpr 3706

csup 7312

cr 8168

cpnf 8347

cmnf 8348

cxr 8349

clt 8350

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-pow 4306 ax-un 4573 ax-setind 4679 ax-cnex 8260 ax-resscn 8261

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-if 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-uni 3931 df-sup 7314 df-pnf 8352 df-mnf 8353 df-xr 8354

This theorem is referenced by: xrmaxiflemval 11994

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