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Theorem xrmaxiflemcom 11176

Description: Lemma for xrmaxif 11178. 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 3522	. . 3 $/\$ $/\$
3		xrpnfdc 9769	. . . . . . 7 DECID
4	3	ad2antrr 480	. . . . . 6 $/\$ $/\$ DECID
5		exmiddc 826	. . . . . 6 DECID $\/$
6	4, 5	syl 14	. . . . 5 $/\$ $/\$ $\/$
7		eqid 2164	. . . . . . . 8
8	7	biantru 300	. . . . . . 7 $/\$
9	8	a1i 9	. . . . . 6 $/\$ $/\$ $/\$
10		xrmnfdc 9770	. . . . . . . . . . 11 DECID
11	10	ad2antrr 480	. . . . . . . . . 10 $/\$ $/\$ DECID
12		exmiddc 826	. . . . . . . . . 10 DECID $\/$
13	11, 12	syl 14	. . . . . . . . 9 $/\$ $/\$ $\/$
14		iba 298	. . . . . . . . . . . 12 $/\$
15	14	eqcoms 2167	. . . . . . . . . . 11 $/\$
16	15	adantl 275	. . . . . . . . . 10 $/\$ $/\$ $/\$
17	1	iftrued 3522	. . . . . . . . . . . 12 $/\$ $/\$
18	17	eqcomd 2170	. . . . . . . . . . 11 $/\$ $/\$
19	18	biantrud 302	. . . . . . . . . 10 $/\$ $/\$ $/\$
20	16, 19	orbi12d 783	. . . . . . . . 9 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
21	13, 20	mpbid 146	. . . . . . . 8 $/\$ $/\$ $/\$ $\/$ $/\$
22		eqifdc 3549	. . . . . . . . 9 DECID $/\$ $\/$ $/\$
23	11, 22	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$ $\/$ $/\$
24	21, 23	mpbird 166	. . . . . . 7 $/\$ $/\$
25	24	biantrud 302	. . . . . 6 $/\$ $/\$ $/\$
26	9, 25	orbi12d 783	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
27	6, 26	mpbid 146	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
28		eqifdc 3549	. . . . 5 DECID $/\$ $\/$ $/\$
29	4, 28	syl 14	. . . 4 $/\$ $/\$ $/\$ $\/$ $/\$
30	27, 29	mpbird 166	. . 3 $/\$ $/\$
31	2, 30	eqtrd 2197	. 2 $/\$ $/\$
32	3, 5	syl 14	. . . . . . . 8 $\/$
33	32	ad3antrrr 484	. . . . . . 7 $/\$ $/\$ $/\$ $\/$
34		pm4.24 393	. . . . . . . . 9 $/\$
35	34	a1i 9	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
36	10	ad3antrrr 484	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ DECID
37	36, 12	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $\/$
38		simpr 109	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
39	38	eqeq2d 2176	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
40	39	anbi2d 460	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
41		anidm 394	. . . . . . . . . . . . 13 $/\$
42	40, 41	bitr2di 196	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
43		eqid 2164	. . . . . . . . . . . . . 14
44	43	biantru 300	. . . . . . . . . . . . 13 $/\$
45	44	a1i 9	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
46	42, 45	orbi12d 783	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
47	37, 46	mpbid 146	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
48		eqifdc 3549	. . . . . . . . . . 11 DECID $/\$ $\/$ $/\$
49	36, 48	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
50	47, 49	mpbird 166	. . . . . . . . 9 $/\$ $/\$ $/\$
51	50	biantrud 302	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
52	35, 51	orbi12d 783	. . . . . . 7 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
53	33, 52	mpbid 146	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
54	3	ad3antrrr 484	. . . . . . 7 $/\$ $/\$ $/\$ DECID
55		eqifdc 3549	. . . . . . 7 DECID $/\$ $\/$ $/\$
56	54, 55	syl 14	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
57	53, 56	mpbird 166	. . . . 5 $/\$ $/\$ $/\$
58	38	iftrued 3522	. . . . 5 $/\$ $/\$ $/\$
59	38	iftrued 3522	. . . . . . 7 $/\$ $/\$ $/\$
60	59	ifeq2d 3533	. . . . . 6 $/\$ $/\$ $/\$
61	60	ifeq2d 3533	. . . . 5 $/\$ $/\$ $/\$
62	57, 58, 61	3eqtr4d 2207	. . . 4 $/\$ $/\$ $/\$
63		simpr 109	. . . . . 6 $/\$ $/\$ $/\$
64	63	iffalsed 3525	. . . . 5 $/\$ $/\$ $/\$
65	63	iffalsed 3525	. . . . . . . 8 $/\$ $/\$ $/\$
66	65	ifeq2d 3533	. . . . . . 7 $/\$ $/\$ $/\$
67	66	ifeq2d 3533	. . . . . 6 $/\$ $/\$ $/\$
68		maxcom 11131	. . . . . . 7
69		ifeq2 3519	. . . . . . 7
70		ifeq2 3519	. . . . . . 7
71	68, 69, 70	mp2b 8	. . . . . 6
72	67, 71	eqtrdi 2213	. . . . 5 $/\$ $/\$ $/\$
73	64, 72	eqtr4d 2200	. . . 4 $/\$ $/\$ $/\$
74		xrmnfdc 9770	. . . . . 6 DECID
75		exmiddc 826	. . . . . 6 DECID $\/$
76	74, 75	syl 14	. . . . 5 $\/$
77	76	ad2antlr 481	. . . 4 $/\$ $/\$ $\/$
78	62, 73, 77	mpjaodan 788	. . 3 $/\$ $/\$
79		simpr 109	. . . 4 $/\$ $/\$
80	79	iffalsed 3525	. . 3 $/\$ $/\$
81	79	iffalsed 3525	. . . . 5 $/\$ $/\$
82	81	ifeq2d 3533	. . . 4 $/\$ $/\$
83	82	ifeq2d 3533	. . 3 $/\$ $/\$
84	78, 80, 83	3eqtr4d 2207	. 2 $/\$ $/\$
85		xrpnfdc 9769	. . . 4 DECID
86		exmiddc 826	. . . 4 DECID $\/$
87	85, 86	syl 14	. . 3 $\/$
88	87	adantl 275	. 2 $/\$ $\/$
89	31, 84, 88	mpjaodan 788	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 698 DECID wdc 824

wceq 1342

wcel 2135

cif 3515

cpr 3571

csup 6938

cr 7743

cpnf 7921

cmnf 7922

cxr 7923

clt 7924

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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-uni 3784 df-sup 6940 df-pnf 7926 df-mnf 7927 df-xr 7928

This theorem is referenced by: xrmaxiflemval 11177

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