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Mirrors > Home > MPE Home > Th. List > mircom | Structured version Visualization version GIF version |
Description: Variation on mirmir 27880. (Contributed by Thierry Arnoux, 10-Nov-2019.) |
Ref | Expression |
---|---|
mirval.p | ⊢ 𝑃 = (Base‘𝐺) |
mirval.d | ⊢ − = (dist‘𝐺) |
mirval.i | ⊢ 𝐼 = (Itv‘𝐺) |
mirval.l | ⊢ 𝐿 = (LineG‘𝐺) |
mirval.s | ⊢ 𝑆 = (pInvG‘𝐺) |
mirval.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
mirval.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
mirfv.m | ⊢ 𝑀 = (𝑆‘𝐴) |
mirmir.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
mircom.1 | ⊢ (𝜑 → (𝑀‘𝐵) = 𝐶) |
Ref | Expression |
---|---|
mircom | ⊢ (𝜑 → (𝑀‘𝐶) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mircom.1 | . . 3 ⊢ (𝜑 → (𝑀‘𝐵) = 𝐶) | |
2 | 1 | fveq2d 6885 | . 2 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐵)) = (𝑀‘𝐶)) |
3 | mirval.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
4 | mirval.d | . . 3 ⊢ − = (dist‘𝐺) | |
5 | mirval.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
6 | mirval.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
7 | mirval.s | . . 3 ⊢ 𝑆 = (pInvG‘𝐺) | |
8 | mirval.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
9 | mirval.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
10 | mirfv.m | . . 3 ⊢ 𝑀 = (𝑆‘𝐴) | |
11 | mirmir.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
12 | 3, 4, 5, 6, 7, 8, 9, 10, 11 | mirmir 27880 | . 2 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐵)) = 𝐵) |
13 | 2, 12 | eqtr3d 2775 | 1 ⊢ (𝜑 → (𝑀‘𝐶) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ‘cfv 6535 Basecbs 17131 distcds 17193 TarskiGcstrkg 27645 Itvcitv 27651 LineGclng 27652 pInvGcmir 27870 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pr 5423 |
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-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-trkgc 27666 df-trkgb 27667 df-trkgcb 27668 df-trkg 27671 df-mir 27871 |
This theorem is referenced by: miduniq 27903 colperpexlem3 27950 mideulem2 27952 midex 27955 opphllem1 27965 opphllem2 27966 opphllem3 27967 opphllem5 27969 opphllem6 27970 trgcopyeulem 28023 |
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