giclcl - Metamath Proof Explorer

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Theorem giclcl 19240

Description: Isomorphism implies the left side is a group. (Contributed by Stefan O'Rear, 25-Jan-2015.)

**Assertion**
Ref	Expression
giclcl	⊢ (𝑅 ≃_𝑔 𝑆 → 𝑅 ∈ Grp)

Proof of Theorem giclcl Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brgic 19237	. . 3 ⊢ (𝑅 ≃_𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅)
2		n0 4282	. . 3 ⊢ ((𝑅 GrpIso 𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆))
3	1, 2	bitri 276	. 2 ⊢ (𝑅 ≃_𝑔 𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆))
4		gimghm 19231	. . . 4 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝑓 ∈ (𝑅 GrpHom 𝑆))
5		ghmgrp1 19185	. . . 4 ⊢ (𝑓 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp)
6	4, 5	syl 17	. . 3 ⊢ (𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝑅 ∈ Grp)
7	6	exlimiv 1937	. 2 ⊢ (∃𝑓 𝑓 ∈ (𝑅 GrpIso 𝑆) → 𝑅 ∈ Grp)
8	3, 7	sylbi 218	1 ⊢ (𝑅 ≃_𝑔 𝑆 → 𝑅 ∈ Grp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∃wex 1786 ∈ wcel 2119 ≠ wne 2934 ∅c0 4262 class class class wbr 5073 (class class class)co 7357 Grpcgrp 18901 GrpHom cghm 19179 GrpIso cgim 19224 ≃_𝑔 cgic 19225

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4840 df-iun 4924 df-br 5074 df-opab 5136 df-mpt 5155 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-ov 7360 df-oprab 7361 df-mpo 7362 df-1st 7932 df-2nd 7933 df-1o 8396 df-map 8766 df-ghm 19180 df-gim 19226 df-gic 19227

This theorem is referenced by: gicer 19244

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