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Mirrors > Home > MPE Home > Th. List > f1otrgds | Structured version Visualization version GIF version |
Description: Convenient lemma for f1otrg 28747. (Contributed by Thierry Arnoux, 19-Mar-2019.) |
Ref | Expression |
---|---|
f1otrkg.p | ⊢ 𝑃 = (Base‘𝐺) |
f1otrkg.d | ⊢ 𝐷 = (dist‘𝐺) |
f1otrkg.i | ⊢ 𝐼 = (Itv‘𝐺) |
f1otrkg.b | ⊢ 𝐵 = (Base‘𝐻) |
f1otrkg.e | ⊢ 𝐸 = (dist‘𝐻) |
f1otrkg.j | ⊢ 𝐽 = (Itv‘𝐻) |
f1otrkg.f | ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝑃) |
f1otrkg.1 | ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵)) → (𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓))) |
f1otrkg.2 | ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑒)𝐼(𝐹‘𝑓)))) |
f1otrgitv.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
f1otrgitv.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
f1otrgds | ⊢ (𝜑 → (𝑋𝐸𝑌) = ((𝐹‘𝑋)𝐷(𝐹‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1otrkg.1 | . . 3 ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵)) → (𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓))) | |
2 | 1 | ralrimivva 3190 | . 2 ⊢ (𝜑 → ∀𝑒 ∈ 𝐵 ∀𝑓 ∈ 𝐵 (𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓))) |
3 | f1otrgitv.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
4 | f1otrgitv.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
5 | oveq1 7426 | . . . . 5 ⊢ (𝑒 = 𝑋 → (𝑒𝐸𝑓) = (𝑋𝐸𝑓)) | |
6 | fveq2 6896 | . . . . . 6 ⊢ (𝑒 = 𝑋 → (𝐹‘𝑒) = (𝐹‘𝑋)) | |
7 | 6 | oveq1d 7434 | . . . . 5 ⊢ (𝑒 = 𝑋 → ((𝐹‘𝑒)𝐷(𝐹‘𝑓)) = ((𝐹‘𝑋)𝐷(𝐹‘𝑓))) |
8 | 5, 7 | eqeq12d 2741 | . . . 4 ⊢ (𝑒 = 𝑋 → ((𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓)) ↔ (𝑋𝐸𝑓) = ((𝐹‘𝑋)𝐷(𝐹‘𝑓)))) |
9 | oveq2 7427 | . . . . 5 ⊢ (𝑓 = 𝑌 → (𝑋𝐸𝑓) = (𝑋𝐸𝑌)) | |
10 | fveq2 6896 | . . . . . 6 ⊢ (𝑓 = 𝑌 → (𝐹‘𝑓) = (𝐹‘𝑌)) | |
11 | 10 | oveq2d 7435 | . . . . 5 ⊢ (𝑓 = 𝑌 → ((𝐹‘𝑋)𝐷(𝐹‘𝑓)) = ((𝐹‘𝑋)𝐷(𝐹‘𝑌))) |
12 | 9, 11 | eqeq12d 2741 | . . . 4 ⊢ (𝑓 = 𝑌 → ((𝑋𝐸𝑓) = ((𝐹‘𝑋)𝐷(𝐹‘𝑓)) ↔ (𝑋𝐸𝑌) = ((𝐹‘𝑋)𝐷(𝐹‘𝑌)))) |
13 | 8, 12 | rspc2v 3617 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑒 ∈ 𝐵 ∀𝑓 ∈ 𝐵 (𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓)) → (𝑋𝐸𝑌) = ((𝐹‘𝑋)𝐷(𝐹‘𝑌)))) |
14 | 3, 4, 13 | syl2anc 582 | . 2 ⊢ (𝜑 → (∀𝑒 ∈ 𝐵 ∀𝑓 ∈ 𝐵 (𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓)) → (𝑋𝐸𝑌) = ((𝐹‘𝑋)𝐷(𝐹‘𝑌)))) |
15 | 2, 14 | mpd 15 | 1 ⊢ (𝜑 → (𝑋𝐸𝑌) = ((𝐹‘𝑋)𝐷(𝐹‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3050 –1-1-onto→wf1o 6548 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 distcds 17245 Itvcitv 28309 |
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 2696 |
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 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-iota 6501 df-fv 6557 df-ov 7422 |
This theorem is referenced by: f1otrg 28747 |
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