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Mirrors > Home > MPE Home > Th. List > climmpt | Structured version Visualization version Unicode version |
Description: Exhibit a function ![]() ![]() |
Ref | Expression |
---|---|
2clim.1 |
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climmpt.2 |
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Ref | Expression |
---|---|
climmpt |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2clim.1 |
. 2
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2 | simpr 463 |
. 2
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3 | climmpt.2 |
. . . 4
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4 | fvex 5873 |
. . . . . 6
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5 | 1, 4 | eqeltri 2524 |
. . . . 5
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6 | 5 | mptex 6134 |
. . . 4
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7 | 3, 6 | eqeltri 2524 |
. . 3
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8 | 7 | a1i 11 |
. 2
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9 | simpl 459 |
. 2
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10 | fveq2 5863 |
. . . . 5
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11 | fvex 5873 |
. . . . 5
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12 | 10, 3, 11 | fvmpt 5946 |
. . . 4
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13 | 12 | eqcomd 2456 |
. . 3
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14 | 13 | adantl 468 |
. 2
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15 | 1, 2, 8, 9, 14 | climeq 13624 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-rep 4514 ax-sep 4524 ax-nul 4533 ax-pow 4580 ax-pr 4638 ax-un 6580 ax-cnex 9592 ax-resscn 9593 ax-pre-lttri 9610 ax-pre-lttrn 9611 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 985 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-nel 2624 df-ral 2741 df-rex 2742 df-reu 2743 df-rab 2745 df-v 3046 df-sbc 3267 df-csb 3363 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-nul 3731 df-if 3881 df-pw 3952 df-sn 3968 df-pr 3970 df-op 3974 df-uni 4198 df-iun 4279 df-br 4402 df-opab 4461 df-mpt 4462 df-id 4748 df-po 4754 df-so 4755 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-ima 4846 df-iota 5545 df-fun 5583 df-fn 5584 df-f 5585 df-f1 5586 df-fo 5587 df-f1o 5588 df-fv 5589 df-ov 6291 df-er 7360 df-en 7567 df-dom 7568 df-sdom 7569 df-pnf 9674 df-mnf 9675 df-xr 9676 df-ltxr 9677 df-le 9678 df-neg 9860 df-z 10935 df-uz 11157 df-clim 13545 |
This theorem is referenced by: climmpt2 13630 climrecl 13640 climge0 13641 caurcvg2 13737 caucvg 13738 climfsum 13873 dstfrvclim1 29303 divcnvg 37701 |
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