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Mirrors > Home > MPE Home > Th. List > f1otrgds | Structured version Visualization version GIF version |
Description: Convenient lemma for f1otrg 27855. (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 7369 | . . . . 5 β’ (π = π β (ππΈπ) = (ππΈπ)) | |
6 | fveq2 6847 | . . . . . 6 β’ (π = π β (πΉβπ) = (πΉβπ)) | |
7 | 6 | oveq1d 7377 | . . . . 5 β’ (π = π β ((πΉβπ)π·(πΉβπ)) = ((πΉβπ)π·(πΉβπ))) |
8 | 5, 7 | eqeq12d 2753 | . . . 4 β’ (π = π β ((ππΈπ) = ((πΉβπ)π·(πΉβπ)) β (ππΈπ) = ((πΉβπ)π·(πΉβπ)))) |
9 | oveq2 7370 | . . . . 5 β’ (π = π β (ππΈπ) = (ππΈπ)) | |
10 | fveq2 6847 | . . . . . 6 β’ (π = π β (πΉβπ) = (πΉβπ)) | |
11 | 10 | oveq2d 7378 | . . . . 5 β’ (π = π β ((πΉβπ)π·(πΉβπ)) = ((πΉβπ)π·(πΉβπ))) |
12 | 9, 11 | eqeq12d 2753 | . . . 4 β’ (π = π β ((ππΈπ) = ((πΉβπ)π·(πΉβπ)) β (ππΈπ) = ((πΉβπ)π·(πΉβπ)))) |
13 | 8, 12 | rspc2v 3593 | . . 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 585 | . 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3065 β1-1-ontoβwf1o 6500 βcfv 6501 (class class class)co 7362 Basecbs 17090 distcds 17149 Itvcitv 27417 |
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-ral 3066 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-iota 6453 df-fv 6509 df-ov 7365 |
This theorem is referenced by: f1otrg 27855 |
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