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Mirrors > Home > MPE Home > Th. List > f1otrgds | Structured version Visualization version GIF version |
Description: Convenient lemma for f1otrg 28630. (Contributed by Thierry Arnoux, 19-Mar-2019.) |
Ref | Expression |
---|---|
f1otrkg.p | β’ π = (BaseβπΊ) |
f1otrkg.d | β’ π· = (distβπΊ) |
f1otrkg.i | β’ πΌ = (ItvβπΊ) |
f1otrkg.b | β’ π΅ = (Baseβπ») |
f1otrkg.e | β’ πΈ = (distβπ») |
f1otrkg.j | β’ π½ = (Itvβπ») |
f1otrkg.f | β’ (π β πΉ:π΅β1-1-ontoβπ) |
f1otrkg.1 | β’ ((π β§ (π β π΅ β§ π β π΅)) β (ππΈπ) = ((πΉβπ)π·(πΉβπ))) |
f1otrkg.2 | β’ ((π β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (π β (ππ½π) β (πΉβπ) β ((πΉβπ)πΌ(πΉβπ)))) |
f1otrgitv.x | β’ (π β π β π΅) |
f1otrgitv.y | β’ (π β π β π΅) |
Ref | Expression |
---|---|
f1otrgds | β’ (π β (ππΈπ) = ((πΉβπ)π·(πΉβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1otrkg.1 | . . 3 β’ ((π β§ (π β π΅ β§ π β π΅)) β (ππΈπ) = ((πΉβπ)π·(πΉβπ))) | |
2 | 1 | ralrimivva 3194 | . 2 β’ (π β βπ β π΅ βπ β π΅ (ππΈπ) = ((πΉβπ)π·(πΉβπ))) |
3 | f1otrgitv.x | . . 3 β’ (π β π β π΅) | |
4 | f1otrgitv.y | . . 3 β’ (π β π β π΅) | |
5 | oveq1 7412 | . . . . 5 β’ (π = π β (ππΈπ) = (ππΈπ)) | |
6 | fveq2 6885 | . . . . . 6 β’ (π = π β (πΉβπ) = (πΉβπ)) | |
7 | 6 | oveq1d 7420 | . . . . 5 β’ (π = π β ((πΉβπ)π·(πΉβπ)) = ((πΉβπ)π·(πΉβπ))) |
8 | 5, 7 | eqeq12d 2742 | . . . 4 β’ (π = π β ((ππΈπ) = ((πΉβπ)π·(πΉβπ)) β (ππΈπ) = ((πΉβπ)π·(πΉβπ)))) |
9 | oveq2 7413 | . . . . 5 β’ (π = π β (ππΈπ) = (ππΈπ)) | |
10 | fveq2 6885 | . . . . . 6 β’ (π = π β (πΉβπ) = (πΉβπ)) | |
11 | 10 | oveq2d 7421 | . . . . 5 β’ (π = π β ((πΉβπ)π·(πΉβπ)) = ((πΉβπ)π·(πΉβπ))) |
12 | 9, 11 | eqeq12d 2742 | . . . 4 β’ (π = π β ((ππΈπ) = ((πΉβπ)π·(πΉβπ)) β (ππΈπ) = ((πΉβπ)π·(πΉβπ)))) |
13 | 8, 12 | rspc2v 3617 | . . 3 β’ ((π β π΅ β§ π β π΅) β (βπ β π΅ βπ β π΅ (ππΈπ) = ((πΉβπ)π·(πΉβπ)) β (ππΈπ) = ((πΉβπ)π·(πΉβπ)))) |
14 | 3, 4, 13 | syl2anc 583 | . 2 β’ (π β (βπ β π΅ βπ β π΅ (ππΈπ) = ((πΉβπ)π·(πΉβπ)) β (ππΈπ) = ((πΉβπ)π·(πΉβπ)))) |
15 | 2, 14 | mpd 15 | 1 β’ (π β (ππΈπ) = ((πΉβπ)π·(πΉβπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3055 β1-1-ontoβwf1o 6536 βcfv 6537 (class class class)co 7405 Basecbs 17153 distcds 17215 Itvcitv 28192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-iota 6489 df-fv 6545 df-ov 7408 |
This theorem is referenced by: f1otrg 28630 |
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