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Mirrors > Home > MPE Home > Th. List > f1otrgds | Structured version Visualization version GIF version |
Description: Convenient lemma for f1otrg 28703. (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 3198 | . 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 7433 | . . . . 5 β’ (π = π β (ππΈπ) = (ππΈπ)) | |
6 | fveq2 6902 | . . . . . 6 β’ (π = π β (πΉβπ) = (πΉβπ)) | |
7 | 6 | oveq1d 7441 | . . . . 5 β’ (π = π β ((πΉβπ)π·(πΉβπ)) = ((πΉβπ)π·(πΉβπ))) |
8 | 5, 7 | eqeq12d 2744 | . . . 4 β’ (π = π β ((ππΈπ) = ((πΉβπ)π·(πΉβπ)) β (ππΈπ) = ((πΉβπ)π·(πΉβπ)))) |
9 | oveq2 7434 | . . . . 5 β’ (π = π β (ππΈπ) = (ππΈπ)) | |
10 | fveq2 6902 | . . . . . 6 β’ (π = π β (πΉβπ) = (πΉβπ)) | |
11 | 10 | oveq2d 7442 | . . . . 5 β’ (π = π β ((πΉβπ)π·(πΉβπ)) = ((πΉβπ)π·(πΉβπ))) |
12 | 9, 11 | eqeq12d 2744 | . . . 4 β’ (π = π β ((ππΈπ) = ((πΉβπ)π·(πΉβπ)) β (ππΈπ) = ((πΉβπ)π·(πΉβπ)))) |
13 | 8, 12 | rspc2v 3622 | . . 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 582 | . 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 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3058 β1-1-ontoβwf1o 6552 βcfv 6553 (class class class)co 7426 Basecbs 17189 distcds 17251 Itvcitv 28265 |
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 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-iota 6505 df-fv 6561 df-ov 7429 |
This theorem is referenced by: f1otrg 28703 |
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