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Mirrors > Home > MPE Home > Th. List > gictr | Structured version Visualization version GIF version |
Description: Isomorphism is transitive. (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
gictr | ⊢ ((𝑅 ≃𝑔 𝑆 ∧ 𝑆 ≃𝑔 𝑇) → 𝑅 ≃𝑔 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brgic 17711 | . 2 ⊢ (𝑅 ≃𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅) | |
2 | brgic 17711 | . 2 ⊢ (𝑆 ≃𝑔 𝑇 ↔ (𝑆 GrpIso 𝑇) ≠ ∅) | |
3 | n0 3931 | . . 3 ⊢ ((𝑅 GrpIso 𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆)) | |
4 | n0 3931 | . . 3 ⊢ ((𝑆 GrpIso 𝑇) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (𝑆 GrpIso 𝑇)) | |
5 | eeanv 2182 | . . . 4 ⊢ (∃𝑓∃𝑔(𝑓 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑔 ∈ (𝑆 GrpIso 𝑇)) ↔ (∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆) ∧ ∃𝑔 𝑔 ∈ (𝑆 GrpIso 𝑇))) | |
6 | gimco 17710 | . . . . . . 7 ⊢ ((𝑔 ∈ (𝑆 GrpIso 𝑇) ∧ 𝑓 ∈ (𝑅 GrpIso 𝑆)) → (𝑔 ∘ 𝑓) ∈ (𝑅 GrpIso 𝑇)) | |
7 | brgici 17712 | . . . . . . 7 ⊢ ((𝑔 ∘ 𝑓) ∈ (𝑅 GrpIso 𝑇) → 𝑅 ≃𝑔 𝑇) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ ((𝑔 ∈ (𝑆 GrpIso 𝑇) ∧ 𝑓 ∈ (𝑅 GrpIso 𝑆)) → 𝑅 ≃𝑔 𝑇) |
9 | 8 | ancoms 469 | . . . . 5 ⊢ ((𝑓 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑔 ∈ (𝑆 GrpIso 𝑇)) → 𝑅 ≃𝑔 𝑇) |
10 | 9 | exlimivv 1860 | . . . 4 ⊢ (∃𝑓∃𝑔(𝑓 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑔 ∈ (𝑆 GrpIso 𝑇)) → 𝑅 ≃𝑔 𝑇) |
11 | 5, 10 | sylbir 225 | . . 3 ⊢ ((∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆) ∧ ∃𝑔 𝑔 ∈ (𝑆 GrpIso 𝑇)) → 𝑅 ≃𝑔 𝑇) |
12 | 3, 4, 11 | syl2anb 496 | . 2 ⊢ (((𝑅 GrpIso 𝑆) ≠ ∅ ∧ (𝑆 GrpIso 𝑇) ≠ ∅) → 𝑅 ≃𝑔 𝑇) |
13 | 1, 2, 12 | syl2anb 496 | 1 ⊢ ((𝑅 ≃𝑔 𝑆 ∧ 𝑆 ≃𝑔 𝑇) → 𝑅 ≃𝑔 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∃wex 1704 ∈ wcel 1990 ≠ wne 2794 ∅c0 3915 class class class wbr 4653 ∘ ccom 5118 (class class class)co 6650 GrpIso cgim 17699 ≃𝑔 cgic 17700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-1o 7560 df-map 7859 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-grp 17425 df-ghm 17658 df-gim 17701 df-gic 17702 |
This theorem is referenced by: gicer 17718 gicerOLD 17719 cyggic 19921 |
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