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Mirrors > Home > MPE Home > Th. List > imbrov2fvoveq | Structured version Visualization version GIF version |
Description: Equality theorem for nested function and operation value in an implication for a binary relation. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 17-Aug-2022.) |
Ref | Expression |
---|---|
imbrov2fvoveq.1 | โข (๐ = ๐ โ (๐ โ ๐)) |
Ref | Expression |
---|---|
imbrov2fvoveq | โข (๐ = ๐ โ ((๐ โ (๐นโ((๐บโ๐) ยท ๐))๐ ๐ด) โ (๐ โ (๐นโ((๐บโ๐) ยท ๐))๐ ๐ด))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imbrov2fvoveq.1 | . 2 โข (๐ = ๐ โ (๐ โ ๐)) | |
2 | fveq2 6843 | . . . 4 โข (๐ = ๐ โ (๐บโ๐) = (๐บโ๐)) | |
3 | 2 | fvoveq1d 7380 | . . 3 โข (๐ = ๐ โ (๐นโ((๐บโ๐) ยท ๐)) = (๐นโ((๐บโ๐) ยท ๐))) |
4 | 3 | breq1d 5116 | . 2 โข (๐ = ๐ โ ((๐นโ((๐บโ๐) ยท ๐))๐ ๐ด โ (๐นโ((๐บโ๐) ยท ๐))๐ ๐ด)) |
5 | 1, 4 | imbi12d 345 | 1 โข (๐ = ๐ โ ((๐ โ (๐นโ((๐บโ๐) ยท ๐))๐ ๐ด) โ (๐ โ (๐นโ((๐บโ๐) ยท ๐))๐ ๐ด))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โ wb 205 = wceq 1542 class class class wbr 5106 โcfv 6497 (class class class)co 7358 |
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-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 df-ov 7361 |
This theorem is referenced by: rlim2 15379 rlimclim1 15428 rlimcn1 15471 climcn1 15475 caucvgrlem 15558 cncfco 24273 ftc1lem4 25406 ftc1lem6 25408 itg2gt0cn 36136 ftc1cnnclem 36152 ftc1cnnc 36153 idlimc 43874 limcperiod 43876 limclner 43899 cncfshift 44122 cncfperiod 44127 fperdvper 44167 |
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