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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 6891 | . . . 4 โข (๐ = ๐ โ (๐บโ๐) = (๐บโ๐)) | |
3 | 2 | fvoveq1d 7430 | . . 3 โข (๐ = ๐ โ (๐นโ((๐บโ๐) ยท ๐)) = (๐นโ((๐บโ๐) ยท ๐))) |
4 | 3 | breq1d 5158 | . 2 โข (๐ = ๐ โ ((๐นโ((๐บโ๐) ยท ๐))๐ ๐ด โ (๐นโ((๐บโ๐) ยท ๐))๐ ๐ด)) |
5 | 1, 4 | imbi12d 344 | 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 1541 class class class wbr 5148 โcfv 6543 (class class class)co 7408 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-ov 7411 |
This theorem is referenced by: rlim2 15439 rlimclim1 15488 rlimcn1 15531 climcn1 15535 caucvgrlem 15618 cncfco 24422 ftc1lem4 25555 ftc1lem6 25557 itg2gt0cn 36538 ftc1cnnclem 36554 ftc1cnnc 36555 idlimc 44332 limcperiod 44334 limclner 44357 cncfshift 44580 cncfperiod 44585 fperdvper 44625 |
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