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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 6892 | . . . 4 โข (๐ = ๐ โ (๐บโ๐) = (๐บโ๐)) | |
3 | 2 | fvoveq1d 7431 | . . 3 โข (๐ = ๐ โ (๐นโ((๐บโ๐) ยท ๐)) = (๐นโ((๐บโ๐) ยท ๐))) |
4 | 3 | breq1d 5159 | . 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 5149 โcfv 6544 (class class class)co 7409 |
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 2704 |
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 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-iota 6496 df-fv 6552 df-ov 7412 |
This theorem is referenced by: rlim2 15440 rlimclim1 15489 rlimcn1 15532 climcn1 15536 caucvgrlem 15619 cncfco 24423 ftc1lem4 25556 ftc1lem6 25558 itg2gt0cn 36543 ftc1cnnclem 36559 ftc1cnnc 36560 idlimc 44342 limcperiod 44344 limclner 44367 cncfshift 44590 cncfperiod 44595 fperdvper 44635 |
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