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| Mirrors > Home > MPE Home > Th. List > xrge0subm | Structured version Visualization version GIF version | ||
| Description: The nonnegative extended real numbers are a submonoid of the nonnegative-infinite extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.) |
| Ref | Expression |
|---|---|
| xrs1mnd.1 | ⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) |
| Ref | Expression |
|---|---|
| xrge0subm | ⊢ (0[,]+∞) ∈ (SubMnd‘𝑅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpl 482 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥) → 𝑥 ∈ ℝ*) | |
| 2 | ge0nemnf 13215 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥) → 𝑥 ≠ -∞) | |
| 3 | 1, 2 | jca 511 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥) → (𝑥 ∈ ℝ* ∧ 𝑥 ≠ -∞)) |
| 4 | elxrge0 13497 | . . . 4 ⊢ (𝑥 ∈ (0[,]+∞) ↔ (𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥)) | |
| 5 | eldifsn 4786 | . . . 4 ⊢ (𝑥 ∈ (ℝ* ∖ {-∞}) ↔ (𝑥 ∈ ℝ* ∧ 𝑥 ≠ -∞)) | |
| 6 | 3, 4, 5 | 3imtr4i 292 | . . 3 ⊢ (𝑥 ∈ (0[,]+∞) → 𝑥 ∈ (ℝ* ∖ {-∞})) |
| 7 | 6 | ssriv 3987 | . 2 ⊢ (0[,]+∞) ⊆ (ℝ* ∖ {-∞}) |
| 8 | 0e0iccpnf 13499 | . 2 ⊢ 0 ∈ (0[,]+∞) | |
| 9 | ge0xaddcl 13502 | . . 3 ⊢ ((𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞)) → (𝑥 +𝑒 𝑦) ∈ (0[,]+∞)) | |
| 10 | 9 | rgen2 3199 | . 2 ⊢ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥 +𝑒 𝑦) ∈ (0[,]+∞) |
| 11 | xrs1mnd.1 | . . . 4 ⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) | |
| 12 | 11 | xrs1mnd 21422 | . . 3 ⊢ 𝑅 ∈ Mnd |
| 13 | difss 4136 | . . . . 5 ⊢ (ℝ* ∖ {-∞}) ⊆ ℝ* | |
| 14 | xrsbas 21396 | . . . . . 6 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
| 15 | 11, 14 | ressbas2 17283 | . . . . 5 ⊢ ((ℝ* ∖ {-∞}) ⊆ ℝ* → (ℝ* ∖ {-∞}) = (Base‘𝑅)) |
| 16 | 13, 15 | ax-mp 5 | . . . 4 ⊢ (ℝ* ∖ {-∞}) = (Base‘𝑅) |
| 17 | 11 | xrs10 21423 | . . . 4 ⊢ 0 = (0g‘𝑅) |
| 18 | xrex 13029 | . . . . . 6 ⊢ ℝ* ∈ V | |
| 19 | 18 | difexi 5330 | . . . . 5 ⊢ (ℝ* ∖ {-∞}) ∈ V |
| 20 | xrsadd 21397 | . . . . . 6 ⊢ +𝑒 = (+g‘ℝ*𝑠) | |
| 21 | 11, 20 | ressplusg 17334 | . . . . 5 ⊢ ((ℝ* ∖ {-∞}) ∈ V → +𝑒 = (+g‘𝑅)) |
| 22 | 19, 21 | ax-mp 5 | . . . 4 ⊢ +𝑒 = (+g‘𝑅) |
| 23 | 16, 17, 22 | issubm 18816 | . . 3 ⊢ (𝑅 ∈ Mnd → ((0[,]+∞) ∈ (SubMnd‘𝑅) ↔ ((0[,]+∞) ⊆ (ℝ* ∖ {-∞}) ∧ 0 ∈ (0[,]+∞) ∧ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥 +𝑒 𝑦) ∈ (0[,]+∞)))) |
| 24 | 12, 23 | ax-mp 5 | . 2 ⊢ ((0[,]+∞) ∈ (SubMnd‘𝑅) ↔ ((0[,]+∞) ⊆ (ℝ* ∖ {-∞}) ∧ 0 ∈ (0[,]+∞) ∧ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥 +𝑒 𝑦) ∈ (0[,]+∞))) |
| 25 | 7, 8, 10, 24 | mpbir3an 1342 | 1 ⊢ (0[,]+∞) ∈ (SubMnd‘𝑅) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 Vcvv 3480 ∖ cdif 3948 ⊆ wss 3951 {csn 4626 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 0cc0 11155 +∞cpnf 11292 -∞cmnf 11293 ℝ*cxr 11294 ≤ cle 11296 +𝑒 cxad 13152 [,]cicc 13390 Basecbs 17247 ↾s cress 17274 +gcplusg 17297 ℝ*𝑠cxrs 17545 Mndcmnd 18747 SubMndcsubmnd 18795 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-xadd 13155 df-icc 13394 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-tset 17316 df-ple 17317 df-ds 17319 df-0g 17486 df-xrs 17547 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 |
| This theorem is referenced by: xrge0cmn 21426 xrge0gsumle 24855 xrge0tsms 24856 xrge0tsmsd 33065 |
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