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Mirrors > Home > MPE Home > Th. List > mircom | Structured version Visualization version GIF version |
Description: Variation on mirmir 26450. (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 6676 | . 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 26450 | . 2 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐵)) = 𝐵) |
13 | 2, 12 | eqtr3d 2860 | 1 ⊢ (𝜑 → (𝑀‘𝐶) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 Basecbs 16485 distcds 16576 TarskiGcstrkg 26218 Itvcitv 26224 LineGclng 26225 pInvGcmir 26440 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-trkgc 26236 df-trkgb 26237 df-trkgcb 26238 df-trkg 26241 df-mir 26441 |
This theorem is referenced by: miduniq 26473 colperpexlem3 26520 mideulem2 26522 midex 26525 opphllem1 26535 opphllem2 26536 opphllem3 26537 opphllem5 26539 opphllem6 26540 trgcopyeulem 26593 |
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