Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > grpofo | Structured version Visualization version GIF version |
Description: A group operation maps onto the group's underlying set. (Contributed by NM, 30-Oct-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grpfo.1 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
grpofo | ⊢ (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)–onto→𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpfo.1 | . . . . . 6 ⊢ 𝑋 = ran 𝐺 | |
2 | 1 | isgrpo 26735 | . . . . 5 ⊢ (𝐺 ∈ GrpOp → (𝐺 ∈ GrpOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢)))) |
3 | 2 | ibi 255 | . . . 4 ⊢ (𝐺 ∈ GrpOp → (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢))) |
4 | 3 | simp1d 1066 | . . 3 ⊢ (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)⟶𝑋) |
5 | 1 | eqcomi 2619 | . . 3 ⊢ ran 𝐺 = 𝑋 |
6 | 4, 5 | jctir 559 | . 2 ⊢ (𝐺 ∈ GrpOp → (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ran 𝐺 = 𝑋)) |
7 | dffo2 6032 | . 2 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ran 𝐺 = 𝑋)) | |
8 | 6, 7 | sylibr 223 | 1 ⊢ (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)–onto→𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 × cxp 5036 ran crn 5039 ⟶wf 5800 –onto→wfo 5802 (class class class)co 6549 GrpOpcgr 26727 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-ov 6552 df-grpo 26731 |
This theorem is referenced by: grpocl 26738 grporndm 26748 grporn 26759 nvgf 26857 hhssabloilem 27502 rngosn3 32893 rngodm1dm2 32901 |
Copyright terms: Public domain | W3C validator |