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Metamath Proof Explorer

Theorem lssnle

Description: Equivalent expressions for "not less than". ( chnlei analog.) (Contributed by NM, 10-Jan-2015) (Revised by Mario Carneiro, 19-Apr-2016)

Ref Expression
Hypotheses lssnle.p ⊢ ⊕ = ( LSSum ‘ 𝐺 )
lssnle.t ⊢ ( 𝜑 → 𝑇 ∈ ( SubGrp ‘ 𝐺 ) )
lssnle.u ⊢ ( 𝜑 → 𝑈 ∈ ( SubGrp ‘ 𝐺 ) )
Assertion lssnle ( 𝜑 → ( ¬ 𝑈 ⊆ 𝑇 ↔ 𝑇 ⊊ ( 𝑇 ⊕ 𝑈 ) ) )

Proof

Step Hyp Ref Expression
1 lssnle.p ⊢ ⊕ = ( LSSum ‘ 𝐺 )
2 lssnle.t ⊢ ( 𝜑 → 𝑇 ∈ ( SubGrp ‘ 𝐺 ) )
3 lssnle.u ⊢ ( 𝜑 → 𝑈 ∈ ( SubGrp ‘ 𝐺 ) )
4 1 lsmss2b ⊢ ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → ( 𝑈 ⊆ 𝑇 ↔ ( 𝑇 ⊕ 𝑈 ) = 𝑇 ) )
5 2 3 4 syl2anc ⊢ ( 𝜑 → ( 𝑈 ⊆ 𝑇 ↔ ( 𝑇 ⊕ 𝑈 ) = 𝑇 ) )
6 eqcom ⊢ ( ( 𝑇 ⊕ 𝑈 ) = 𝑇 ↔ 𝑇 = ( 𝑇 ⊕ 𝑈 ) )
7 5 6 bitrdi ⊢ ( 𝜑 → ( 𝑈 ⊆ 𝑇 ↔ 𝑇 = ( 𝑇 ⊕ 𝑈 ) ) )
8 7 necon3bbid ⊢ ( 𝜑 → ( ¬ 𝑈 ⊆ 𝑇 ↔ 𝑇 ≠ ( 𝑇 ⊕ 𝑈 ) ) )
9 1 lsmub1 ⊢ ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → 𝑇 ⊆ ( 𝑇 ⊕ 𝑈 ) )
10 2 3 9 syl2anc ⊢ ( 𝜑 → 𝑇 ⊆ ( 𝑇 ⊕ 𝑈 ) )
11 df-pss ⊢ ( 𝑇 ⊊ ( 𝑇 ⊕ 𝑈 ) ↔ ( 𝑇 ⊆ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑇 ≠ ( 𝑇 ⊕ 𝑈 ) ) )
12 11 baib ⊢ ( 𝑇 ⊆ ( 𝑇 ⊕ 𝑈 ) → ( 𝑇 ⊊ ( 𝑇 ⊕ 𝑈 ) ↔ 𝑇 ≠ ( 𝑇 ⊕ 𝑈 ) ) )
13 10 12 syl ⊢ ( 𝜑 → ( 𝑇 ⊊ ( 𝑇 ⊕ 𝑈 ) ↔ 𝑇 ≠ ( 𝑇 ⊕ 𝑈 ) ) )
14 8 13 bitr4d ⊢ ( 𝜑 → ( ¬ 𝑈 ⊆ 𝑇 ↔ 𝑇 ⊊ ( 𝑇 ⊕ 𝑈 ) ) )