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Mirrors > Home > MPE Home > Th. List > addsgt0d | Structured version Visualization version GIF version |
Description: The sum of two positive surreals is positive. (Contributed by Scott Fenton, 15-Apr-2025.) |
Ref | Expression |
---|---|
addsgt0d.1 | ⊢ (𝜑 → 𝐴 ∈ No ) |
addsgt0d.2 | ⊢ (𝜑 → 𝐵 ∈ No ) |
addsgt0d.3 | ⊢ (𝜑 → 0s <s 𝐴) |
addsgt0d.4 | ⊢ (𝜑 → 0s <s 𝐵) |
Ref | Expression |
---|---|
addsgt0d | ⊢ (𝜑 → 0s <s (𝐴 +s 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0sno 27746 | . . 3 ⊢ 0s ∈ No | |
2 | addsrid 27868 | . . 3 ⊢ ( 0s ∈ No → ( 0s +s 0s ) = 0s ) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ( 0s +s 0s ) = 0s |
4 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → 0s ∈ No ) |
5 | addsgt0d.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ No ) | |
6 | addsgt0d.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ No ) | |
7 | addsgt0d.3 | . . 3 ⊢ (𝜑 → 0s <s 𝐴) | |
8 | addsgt0d.4 | . . 3 ⊢ (𝜑 → 0s <s 𝐵) | |
9 | 4, 4, 5, 6, 7, 8 | slt2addd 27917 | . 2 ⊢ (𝜑 → ( 0s +s 0s ) <s (𝐴 +s 𝐵)) |
10 | 3, 9 | eqbrtrrid 5178 | 1 ⊢ (𝜑 → 0s <s (𝐴 +s 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 class class class wbr 5142 (class class class)co 7414 No csur 27560 <s cslt 27561 0s c0s 27742 +s cadds 27863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-1o 8480 df-2o 8481 df-nadd 8680 df-no 27563 df-slt 27564 df-bday 27565 df-sle 27665 df-sslt 27701 df-scut 27703 df-0s 27744 df-made 27761 df-old 27762 df-left 27764 df-right 27765 df-norec2 27853 df-adds 27864 |
This theorem is referenced by: nnaddscl 28199 |
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