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Mirrors > Home > MPE Home > Th. List > miduniq2 | Structured version Visualization version GIF version |
Description: If two point inversions commute, they are identical. Theorem 7.19 of [Schwabhauser] p. 52. (Contributed by Thierry Arnoux, 30-Jul-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) |
miduniq2.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
miduniq2.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
miduniq2.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
miduniq2.e | ⊢ (𝜑 → ((𝑆‘𝐴)‘((𝑆‘𝐵)‘𝑋)) = ((𝑆‘𝐵)‘((𝑆‘𝐴)‘𝑋))) |
Ref | Expression |
---|---|
miduniq2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mirval.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
2 | mirval.d | . . . 4 ⊢ − = (dist‘𝐺) | |
3 | mirval.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | mirval.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
5 | mirval.s | . . . 4 ⊢ 𝑆 = (pInvG‘𝐺) | |
6 | mirval.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
7 | miduniq2.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
8 | eqid 2820 | . . . . . 6 ⊢ (𝑆‘𝐵) = (𝑆‘𝐵) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | mirf 26444 | . . . . 5 ⊢ (𝜑 → (𝑆‘𝐵):𝑃⟶𝑃) |
10 | miduniq2.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
11 | 9, 10 | ffvelrnd 6845 | . . . 4 ⊢ (𝜑 → ((𝑆‘𝐵)‘𝐴) ∈ 𝑃) |
12 | miduniq2.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
13 | eqid 2820 | . . . . . 6 ⊢ ((𝑆‘𝐵)‘𝐴) = ((𝑆‘𝐵)‘𝐴) | |
14 | eqid 2820 | . . . . . 6 ⊢ ((𝑆‘𝐵)‘((𝑆‘𝐵)‘𝑋)) = ((𝑆‘𝐵)‘((𝑆‘𝐵)‘𝑋)) | |
15 | eqid 2820 | . . . . . 6 ⊢ ((𝑆‘𝐵)‘((𝑆‘𝐵)‘((𝑆‘𝐴)‘𝑋))) = ((𝑆‘𝐵)‘((𝑆‘𝐵)‘((𝑆‘𝐴)‘𝑋))) | |
16 | 9, 12 | ffvelrnd 6845 | . . . . . 6 ⊢ (𝜑 → ((𝑆‘𝐵)‘𝑋) ∈ 𝑃) |
17 | eqid 2820 | . . . . . . . 8 ⊢ (𝑆‘𝐴) = (𝑆‘𝐴) | |
18 | 1, 2, 3, 4, 5, 6, 10, 17, 12 | mircl 26445 | . . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝐴)‘𝑋) ∈ 𝑃) |
19 | 9, 18 | ffvelrnd 6845 | . . . . . 6 ⊢ (𝜑 → ((𝑆‘𝐵)‘((𝑆‘𝐴)‘𝑋)) ∈ 𝑃) |
20 | miduniq2.e | . . . . . 6 ⊢ (𝜑 → ((𝑆‘𝐴)‘((𝑆‘𝐵)‘𝑋)) = ((𝑆‘𝐵)‘((𝑆‘𝐴)‘𝑋))) | |
21 | 1, 2, 3, 4, 5, 6, 8, 13, 14, 15, 7, 10, 16, 19, 20 | mirauto 26468 | . . . . 5 ⊢ (𝜑 → ((𝑆‘((𝑆‘𝐵)‘𝐴))‘((𝑆‘𝐵)‘((𝑆‘𝐵)‘𝑋))) = ((𝑆‘𝐵)‘((𝑆‘𝐵)‘((𝑆‘𝐴)‘𝑋)))) |
22 | 1, 2, 3, 4, 5, 6, 7, 8, 12 | mirmir 26446 | . . . . . 6 ⊢ (𝜑 → ((𝑆‘𝐵)‘((𝑆‘𝐵)‘𝑋)) = 𝑋) |
23 | 22 | fveq2d 6667 | . . . . 5 ⊢ (𝜑 → ((𝑆‘((𝑆‘𝐵)‘𝐴))‘((𝑆‘𝐵)‘((𝑆‘𝐵)‘𝑋))) = ((𝑆‘((𝑆‘𝐵)‘𝐴))‘𝑋)) |
24 | 1, 2, 3, 4, 5, 6, 7, 8, 18 | mirmir 26446 | . . . . 5 ⊢ (𝜑 → ((𝑆‘𝐵)‘((𝑆‘𝐵)‘((𝑆‘𝐴)‘𝑋))) = ((𝑆‘𝐴)‘𝑋)) |
25 | 21, 23, 24 | 3eqtr3d 2863 | . . . 4 ⊢ (𝜑 → ((𝑆‘((𝑆‘𝐵)‘𝐴))‘𝑋) = ((𝑆‘𝐴)‘𝑋)) |
26 | 1, 2, 3, 4, 5, 6, 11, 10, 12, 25 | miduniq1 26470 | . . 3 ⊢ (𝜑 → ((𝑆‘𝐵)‘𝐴) = 𝐴) |
27 | 1, 2, 3, 4, 5, 6, 7, 8, 10 | mirinv 26450 | . . 3 ⊢ (𝜑 → (((𝑆‘𝐵)‘𝐴) = 𝐴 ↔ 𝐵 = 𝐴)) |
28 | 26, 27 | mpbid 234 | . 2 ⊢ (𝜑 → 𝐵 = 𝐴) |
29 | 28 | eqcomd 2826 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ‘cfv 6348 Basecbs 16478 distcds 16569 TarskiGcstrkg 26214 Itvcitv 26220 LineGclng 26221 pInvGcmir 26436 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-pm 8402 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-dju 9323 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-fz 12890 df-fzo 13031 df-hash 13688 df-word 13859 df-concat 13918 df-s1 13945 df-s2 14205 df-s3 14206 df-trkgc 26232 df-trkgb 26233 df-trkgcb 26234 df-trkg 26237 df-cgrg 26295 df-mir 26437 |
This theorem is referenced by: (None) |
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