mircom - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > mircom		Structured version Visualization version GIF version

Theorem mircom 28487

Description: Variation on mirmir 28486. (Contributed by Thierry Arnoux, 10-Nov-2019.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
mircom	⊢ (𝜑 → (𝑀‘𝐶) = 𝐵)

**Proof of Theorem mircom**
Step	Hyp	Ref	Expression
1		mircom.1	. . 3 ⊢ (𝜑 → (𝑀‘𝐵) = 𝐶)
2	1	fveq2d 6906	. 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 28486	. 2 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐵)) = 𝐵)
13	2, 12	eqtr3d 2770	1 ⊢ (𝜑 → (𝑀‘𝐶) = 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6553 Basecbs 17187 distcds 17249 TarskiGcstrkg 28251 Itvcitv 28257 LineGclng 28258 pInvGcmir 28476

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-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433

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-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-trkgc 28272 df-trkgb 28273 df-trkgcb 28274 df-trkg 28277 df-mir 28477

This theorem is referenced by: miduniq 28509 colperpexlem3 28556 mideulem2 28558 midex 28561 opphllem1 28571 opphllem2 28572 opphllem3 28573 opphllem5 28575 opphllem6 28576 trgcopyeulem 28629

W3C validator

	
		OSZAR »