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Mirrors > Home > ILE Home > Th. List > xrmaxiflemcl | Unicode version |
Description: Lemma for xrmaxif 11207. Closure. (Contributed by Jim Kingdon, 29-Apr-2023.) |
Ref | Expression |
---|---|
xrmaxiflemcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfxr 7965 | . . 3 | |
2 | 1 | a1i 9 | . 2 |
3 | simpl 108 | . . . 4 | |
4 | 3 | ad2antrr 485 | . . 3 |
5 | 1 | a1i 9 | . . . 4 |
6 | simpr 109 | . . . . . 6 | |
7 | 6 | ad4antr 491 | . . . . 5 |
8 | simplr 525 | . . . . . . . 8 | |
9 | simpr 109 | . . . . . . . 8 | |
10 | elxr 9726 | . . . . . . . . . 10 | |
11 | 3, 10 | sylib 121 | . . . . . . . . 9 |
12 | 11 | ad4antr 491 | . . . . . . . 8 |
13 | 8, 9, 12 | ecase23d 1345 | . . . . . . 7 |
14 | simp-4r 537 | . . . . . . . 8 | |
15 | simpllr 529 | . . . . . . . 8 | |
16 | elxr 9726 | . . . . . . . . . 10 | |
17 | 6, 16 | sylib 121 | . . . . . . . . 9 |
18 | 17 | ad4antr 491 | . . . . . . . 8 |
19 | 14, 15, 18 | ecase23d 1345 | . . . . . . 7 |
20 | maxcl 11167 | . . . . . . 7 | |
21 | 13, 19, 20 | syl2anc 409 | . . . . . 6 |
22 | 21 | rexrd 7962 | . . . . 5 |
23 | xrmnfdc 9793 | . . . . . 6 DECID | |
24 | 23 | ad4antr 491 | . . . . 5 DECID |
25 | 7, 22, 24 | ifcldadc 3554 | . . . 4 |
26 | xrpnfdc 9792 | . . . . . 6 DECID | |
27 | 3, 26 | syl 14 | . . . . 5 DECID |
28 | 27 | ad2antrr 485 | . . . 4 DECID |
29 | 5, 25, 28 | ifcldadc 3554 | . . 3 |
30 | xrmnfdc 9793 | . . . 4 DECID | |
31 | 30 | ad2antlr 486 | . . 3 DECID |
32 | 4, 29, 31 | ifcldadc 3554 | . 2 |
33 | xrpnfdc 9792 | . . 3 DECID | |
34 | 6, 33 | syl 14 | . 2 DECID |
35 | 2, 32, 34 | ifcldadc 3554 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 DECID wdc 829 w3o 972 wceq 1348 wcel 2141 cif 3525 cpr 3582 csup 6957 cr 7766 cpnf 7944 cmnf 7945 cxr 7946 clt 7947 |
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-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-mulrcl 7866 ax-addcom 7867 ax-mulcom 7868 ax-addass 7869 ax-mulass 7870 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-1rid 7874 ax-0id 7875 ax-rnegex 7876 ax-precex 7877 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-apti 7882 ax-pre-ltadd 7883 ax-pre-mulgt0 7884 ax-pre-mulext 7885 ax-arch 7886 ax-caucvg 7887 |
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-eu 2022 df-mo 2023 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-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-frec 6368 df-sup 6959 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-reap 8487 df-ap 8494 df-div 8583 df-inn 8872 df-2 8930 df-3 8931 df-4 8932 df-n0 9129 df-z 9206 df-uz 9481 df-rp 9604 df-seqfrec 10395 df-exp 10469 df-cj 10799 df-re 10800 df-im 10801 df-rsqrt 10955 df-abs 10956 |
This theorem is referenced by: xrmaxiflemlub 11204 xrmaxiflemval 11206 xrmaxcl 11208 |
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