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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climinff | Structured version Visualization version GIF version |
Description: A version of climinf 43854 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 15-Sep-2020.) |
Ref | Expression |
---|---|
climinff.1 | β’ β²ππ |
climinff.2 | β’ β²ππΉ |
climinff.3 | β’ π = (β€β₯βπ) |
climinff.4 | β’ (π β π β β€) |
climinff.5 | β’ (π β πΉ:πβΆβ) |
climinff.6 | β’ ((π β§ π β π) β (πΉβ(π + 1)) β€ (πΉβπ)) |
climinff.7 | β’ (π β βπ₯ β β βπ β π π₯ β€ (πΉβπ)) |
Ref | Expression |
---|---|
climinff | β’ (π β πΉ β inf(ran πΉ, β, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climinff.3 | . 2 β’ π = (β€β₯βπ) | |
2 | climinff.4 | . 2 β’ (π β π β β€) | |
3 | climinff.5 | . 2 β’ (π β πΉ:πβΆβ) | |
4 | climinff.1 | . . . . 5 β’ β²ππ | |
5 | nfv 1918 | . . . . 5 β’ β²π π β π | |
6 | 4, 5 | nfan 1903 | . . . 4 β’ β²π(π β§ π β π) |
7 | climinff.2 | . . . . . 6 β’ β²ππΉ | |
8 | nfcv 2908 | . . . . . 6 β’ β²π(π + 1) | |
9 | 7, 8 | nffv 6853 | . . . . 5 β’ β²π(πΉβ(π + 1)) |
10 | nfcv 2908 | . . . . 5 β’ β²π β€ | |
11 | nfcv 2908 | . . . . . 6 β’ β²ππ | |
12 | 7, 11 | nffv 6853 | . . . . 5 β’ β²π(πΉβπ) |
13 | 9, 10, 12 | nfbr 5153 | . . . 4 β’ β²π(πΉβ(π + 1)) β€ (πΉβπ) |
14 | 6, 13 | nfim 1900 | . . 3 β’ β²π((π β§ π β π) β (πΉβ(π + 1)) β€ (πΉβπ)) |
15 | eleq1w 2821 | . . . . 5 β’ (π = π β (π β π β π β π)) | |
16 | 15 | anbi2d 630 | . . . 4 β’ (π = π β ((π β§ π β π) β (π β§ π β π))) |
17 | fvoveq1 7381 | . . . . 5 β’ (π = π β (πΉβ(π + 1)) = (πΉβ(π + 1))) | |
18 | fveq2 6843 | . . . . 5 β’ (π = π β (πΉβπ) = (πΉβπ)) | |
19 | 17, 18 | breq12d 5119 | . . . 4 β’ (π = π β ((πΉβ(π + 1)) β€ (πΉβπ) β (πΉβ(π + 1)) β€ (πΉβπ))) |
20 | 16, 19 | imbi12d 345 | . . 3 β’ (π = π β (((π β§ π β π) β (πΉβ(π + 1)) β€ (πΉβπ)) β ((π β§ π β π) β (πΉβ(π + 1)) β€ (πΉβπ)))) |
21 | climinff.6 | . . 3 β’ ((π β§ π β π) β (πΉβ(π + 1)) β€ (πΉβπ)) | |
22 | 14, 20, 21 | chvarfv 2234 | . 2 β’ ((π β§ π β π) β (πΉβ(π + 1)) β€ (πΉβπ)) |
23 | nfcv 2908 | . . . . 5 β’ β²πβ | |
24 | 5 | nfci 2891 | . . . . . 6 β’ β²ππ |
25 | nfcv 2908 | . . . . . . 7 β’ β²ππ₯ | |
26 | 25, 10, 12 | nfbr 5153 | . . . . . 6 β’ β²π π₯ β€ (πΉβπ) |
27 | 24, 26 | nfralw 3295 | . . . . 5 β’ β²πβπ β π π₯ β€ (πΉβπ) |
28 | 23, 27 | nfrexw 3297 | . . . 4 β’ β²πβπ₯ β β βπ β π π₯ β€ (πΉβπ) |
29 | 4, 28 | nfim 1900 | . . 3 β’ β²π(π β βπ₯ β β βπ β π π₯ β€ (πΉβπ)) |
30 | nfv 1918 | . . . . . . 7 β’ β²π π₯ β€ (πΉβπ) | |
31 | 18 | breq2d 5118 | . . . . . . 7 β’ (π = π β (π₯ β€ (πΉβπ) β π₯ β€ (πΉβπ))) |
32 | 30, 26, 31 | cbvralw 3290 | . . . . . 6 β’ (βπ β π π₯ β€ (πΉβπ) β βπ β π π₯ β€ (πΉβπ)) |
33 | 32 | a1i 11 | . . . . 5 β’ (π = π β (βπ β π π₯ β€ (πΉβπ) β βπ β π π₯ β€ (πΉβπ))) |
34 | 33 | rexbidv 3176 | . . . 4 β’ (π = π β (βπ₯ β β βπ β π π₯ β€ (πΉβπ) β βπ₯ β β βπ β π π₯ β€ (πΉβπ))) |
35 | 34 | imbi2d 341 | . . 3 β’ (π = π β ((π β βπ₯ β β βπ β π π₯ β€ (πΉβπ)) β (π β βπ₯ β β βπ β π π₯ β€ (πΉβπ)))) |
36 | climinff.7 | . . 3 β’ (π β βπ₯ β β βπ β π π₯ β€ (πΉβπ)) | |
37 | 29, 35, 36 | chvarfv 2234 | . 2 β’ (π β βπ₯ β β βπ β π π₯ β€ (πΉβπ)) |
38 | 1, 2, 3, 22, 37 | climinf 43854 | 1 β’ (π β πΉ β inf(ran πΉ, β, < )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β²wnf 1786 β wcel 2107 β²wnfc 2888 βwral 3065 βwrex 3074 class class class wbr 5106 ran crn 5635 βΆwf 6493 βcfv 6497 (class class class)co 7358 infcinf 9378 βcr 11051 1c1 11053 + caddc 11055 < clt 11190 β€ cle 11191 β€cz 12500 β€β₯cuz 12764 β cli 15367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-sup 9379 df-inf 9380 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-n0 12415 df-z 12501 df-uz 12765 df-rp 12917 df-fz 13426 df-seq 13908 df-exp 13969 df-cj 14985 df-re 14986 df-im 14987 df-sqrt 15121 df-abs 15122 df-clim 15371 |
This theorem is referenced by: (None) |
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