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Mirrors > Home > ILE Home > Th. List > Mathboxes > nninfsellemeqinf | Unicode version |
Description: Lemma for nninfsel 13213. (Contributed by Jim Kingdon, 9-Aug-2022.) |
Ref | Expression |
---|---|
nninfsel.e | ℕ∞ |
nninfsel.q | ℕ∞ |
nninfsel.1 |
Ref | Expression |
---|---|
nninfsellemeqinf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nninfsel.e | . . . . . . 7 ℕ∞ | |
2 | 1 | nninfself 13209 | . . . . . 6 ℕ∞ℕ∞ |
3 | 2 | a1i 9 | . . . . 5 ℕ∞ℕ∞ |
4 | nninfsel.q | . . . . 5 ℕ∞ | |
5 | 3, 4 | ffvelrnd 5556 | . . . 4 ℕ∞ |
6 | nninff 13198 | . . . 4 ℕ∞ | |
7 | 5, 6 | syl 14 | . . 3 |
8 | 7 | ffnd 5273 | . 2 |
9 | 1onn 6416 | . . . . 5 | |
10 | fnconstg 5320 | . . . . 5 | |
11 | 9, 10 | ax-mp 5 | . . . 4 |
12 | fconstmpt 4586 | . . . . 5 | |
13 | 12 | fneq1i 5217 | . . . 4 |
14 | 11, 13 | mpbi 144 | . . 3 |
15 | 14 | a1i 9 | . 2 |
16 | elequ2 1691 | . . . . . . . . . 10 | |
17 | 16 | ifbid 3493 | . . . . . . . . 9 |
18 | 17 | mpteq2dv 4019 | . . . . . . . 8 |
19 | 18 | fveq2d 5425 | . . . . . . 7 |
20 | 19 | eqeq1d 2148 | . . . . . 6 |
21 | 4 | adantr 274 | . . . . . . . . 9 ℕ∞ |
22 | nninfsel.1 | . . . . . . . . . 10 | |
23 | 22 | adantr 274 | . . . . . . . . 9 |
24 | simpr 109 | . . . . . . . . 9 | |
25 | 1, 21, 23, 24 | nninfsellemqall 13211 | . . . . . . . 8 |
26 | 25 | ralrimiva 2505 | . . . . . . 7 |
27 | 26 | ad2antrr 479 | . . . . . 6 |
28 | simpr 109 | . . . . . . 7 | |
29 | peano2 4509 | . . . . . . . 8 | |
30 | 29 | ad2antlr 480 | . . . . . . 7 |
31 | elnn 4519 | . . . . . . 7 | |
32 | 28, 30, 31 | syl2anc 408 | . . . . . 6 |
33 | 20, 27, 32 | rspcdva 2794 | . . . . 5 |
34 | 33 | ralrimiva 2505 | . . . 4 |
35 | 34 | iftrued 3481 | . . 3 |
36 | omex 4507 | . . . . . . 7 | |
37 | 36 | mptex 5646 | . . . . . 6 |
38 | 37 | a1i 9 | . . . . 5 |
39 | fveq1 5420 | . . . . . . . . . 10 | |
40 | 39 | eqeq1d 2148 | . . . . . . . . 9 |
41 | 40 | ralbidv 2437 | . . . . . . . 8 |
42 | 41 | ifbid 3493 | . . . . . . 7 |
43 | 42 | mpteq2dv 4019 | . . . . . 6 |
44 | 43, 1 | fvmptg 5497 | . . . . 5 ℕ∞ |
45 | 21, 38, 44 | syl2anc 408 | . . . 4 |
46 | suceq 4324 | . . . . . . 7 | |
47 | 46 | adantl 275 | . . . . . 6 |
48 | 47 | raleqdv 2632 | . . . . 5 |
49 | 48 | ifbid 3493 | . . . 4 |
50 | 35, 9 | eqeltrdi 2230 | . . . 4 |
51 | 45, 49, 24, 50 | fvmptd 5502 | . . 3 |
52 | eqidd 2140 | . . . . . 6 | |
53 | eqid 2139 | . . . . . 6 | |
54 | 52, 53 | fvmptg 5497 | . . . . 5 |
55 | 9, 54 | mpan2 421 | . . . 4 |
56 | 55 | adantl 275 | . . 3 |
57 | 35, 51, 56 | 3eqtr4d 2182 | . 2 |
58 | 8, 15, 57 | eqfnfvd 5521 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wral 2416 cvv 2686 c0 3363 cif 3474 csn 3527 cmpt 3989 csuc 4287 com 4504 cxp 4537 wfn 5118 wf 5119 cfv 5123 (class class class)co 5774 c1o 6306 c2o 6307 cmap 6542 ℕ∞xnninf 7005 |
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 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
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-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1o 6313 df-2o 6314 df-map 6544 df-nninf 7007 |
This theorem is referenced by: nninfsel 13213 |
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