abvgt0 - Metamath Proof Explorer

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Theorem abvgt0 20847

Description: The absolute value of a nonzero number is strictly positive. (Contributed by Mario Carneiro, 8-Sep-2014.)

**Hypotheses**
Ref	Expression
abvf.a	⊢ 𝐴 = (AbsVal‘𝑅)
abvf.b	⊢ 𝐵 = (Base‘𝑅)
abveq0.z	⊢ 0 = (0_g‘𝑅)

**Assertion**
Ref	Expression
abvgt0	⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 0 < (𝐹‘𝑋))

**Proof of Theorem abvgt0**
Step	Hyp	Ref	Expression
1		abvf.a	. . . 4 ⊢ 𝐴 = (AbsVal‘𝑅)
2		abvf.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
3	1, 2	abvcl 20843	. . 3 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ ℝ)
4	3	3adant3 1144	. 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐹‘𝑋) ∈ ℝ)
5	1, 2	abvge0 20844	. . 3 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → 0 ≤ (𝐹‘𝑋))
6	5	3adant3 1144	. 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 0 ≤ (𝐹‘𝑋))
7		abveq0.z	. . 3 ⊢ 0 = (0_g‘𝑅)
8	1, 2, 7	abvne0 20846	. 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐹‘𝑋) ≠ 0)
9	4, 6, 8	ne0gt0d 11315	1 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 0 < (𝐹‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 class class class wbr 5099 ‘cfv 6515 ℝcr 11067 0cc0 11068 < clt 11211 ≤ cle 11212 Basecbs 17226 0_gc0g 17449 AbsValcabv 20835

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7712 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-addrcl 11129 ax-rnegex 11139 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5540 df-po 5553 df-so 5554 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-ov 7393 df-oprab 7394 df-mpo 7395 df-er 8671 df-map 8803 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11213 df-mnf 11214 df-xr 11215 df-ltxr 11216 df-le 11217 df-ico 13350 df-abv 20836

This theorem is referenced by: abvres 20858 abvcxp 27654 ostth2 27676 ostth3 27677

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