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| Description: The converse of a motion is a motion. (Contributed by Thierry Arnoux, 15-Dec-2019.) | 
| Ref | Expression | 
|---|---|
| ismot.p | ⊢ 𝑃 = (Base‘𝐺) | 
| ismot.m | ⊢ − = (dist‘𝐺) | 
| motgrp.1 | ⊢ (𝜑 → 𝐺 ∈ 𝑉) | 
| motco.2 | ⊢ (𝜑 → 𝐹 ∈ (𝐺Ismt𝐺)) | 
| Ref | Expression | 
|---|---|
| cnvmot | ⊢ (𝜑 → ◡𝐹 ∈ (𝐺Ismt𝐺)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ismot.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
| 2 | ismot.m | . . . 4 ⊢ − = (dist‘𝐺) | |
| 3 | motgrp.1 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
| 4 | motco.2 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐺Ismt𝐺)) | |
| 5 | 1, 2, 3, 4 | motf1o 28546 | . . 3 ⊢ (𝜑 → 𝐹:𝑃–1-1-onto→𝑃) | 
| 6 | f1ocnv 6860 | . . 3 ⊢ (𝐹:𝑃–1-1-onto→𝑃 → ◡𝐹:𝑃–1-1-onto→𝑃) | |
| 7 | 5, 6 | syl 17 | . 2 ⊢ (𝜑 → ◡𝐹:𝑃–1-1-onto→𝑃) | 
| 8 | 3 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝐺 ∈ 𝑉) | 
| 9 | f1of 6848 | . . . . . . . 8 ⊢ (◡𝐹:𝑃–1-1-onto→𝑃 → ◡𝐹:𝑃⟶𝑃) | |
| 10 | 7, 9 | syl 17 | . . . . . . 7 ⊢ (𝜑 → ◡𝐹:𝑃⟶𝑃) | 
| 11 | 10 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ◡𝐹:𝑃⟶𝑃) | 
| 12 | simprl 771 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝑎 ∈ 𝑃) | |
| 13 | 11, 12 | ffvelcdmd 7105 | . . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (◡𝐹‘𝑎) ∈ 𝑃) | 
| 14 | simprr 773 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝑏 ∈ 𝑃) | |
| 15 | 11, 14 | ffvelcdmd 7105 | . . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (◡𝐹‘𝑏) ∈ 𝑃) | 
| 16 | 4 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → 𝐹 ∈ (𝐺Ismt𝐺)) | 
| 17 | 1, 2, 8, 13, 15, 16 | motcgr 28544 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((𝐹‘(◡𝐹‘𝑎)) − (𝐹‘(◡𝐹‘𝑏))) = ((◡𝐹‘𝑎) − (◡𝐹‘𝑏))) | 
| 18 | f1ocnvfv2 7297 | . . . . . 6 ⊢ ((𝐹:𝑃–1-1-onto→𝑃 ∧ 𝑎 ∈ 𝑃) → (𝐹‘(◡𝐹‘𝑎)) = 𝑎) | |
| 19 | 5, 12, 18 | syl2an2r 685 | . . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (𝐹‘(◡𝐹‘𝑎)) = 𝑎) | 
| 20 | f1ocnvfv2 7297 | . . . . . 6 ⊢ ((𝐹:𝑃–1-1-onto→𝑃 ∧ 𝑏 ∈ 𝑃) → (𝐹‘(◡𝐹‘𝑏)) = 𝑏) | |
| 21 | 5, 14, 20 | syl2an2r 685 | . . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (𝐹‘(◡𝐹‘𝑏)) = 𝑏) | 
| 22 | 19, 21 | oveq12d 7449 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((𝐹‘(◡𝐹‘𝑎)) − (𝐹‘(◡𝐹‘𝑏))) = (𝑎 − 𝑏)) | 
| 23 | 17, 22 | eqtr3d 2779 | . . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((◡𝐹‘𝑎) − (◡𝐹‘𝑏)) = (𝑎 − 𝑏)) | 
| 24 | 23 | ralrimivva 3202 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((◡𝐹‘𝑎) − (◡𝐹‘𝑏)) = (𝑎 − 𝑏)) | 
| 25 | 1, 2 | ismot 28543 | . . 3 ⊢ (𝐺 ∈ 𝑉 → (◡𝐹 ∈ (𝐺Ismt𝐺) ↔ (◡𝐹:𝑃–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((◡𝐹‘𝑎) − (◡𝐹‘𝑏)) = (𝑎 − 𝑏)))) | 
| 26 | 3, 25 | syl 17 | . 2 ⊢ (𝜑 → (◡𝐹 ∈ (𝐺Ismt𝐺) ↔ (◡𝐹:𝑃–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((◡𝐹‘𝑎) − (◡𝐹‘𝑏)) = (𝑎 − 𝑏)))) | 
| 27 | 7, 24, 26 | mpbir2and 713 | 1 ⊢ (𝜑 → ◡𝐹 ∈ (𝐺Ismt𝐺)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ◡ccnv 5684 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 distcds 17306 Ismtcismt 28540 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-ismt 28541 | 
| This theorem is referenced by: motgrp 28551 | 
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