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| Mirrors > Home > MPE Home > Th. List > subsfo | Structured version Visualization version GIF version | ||
| Description: Surreal subtraction is an onto function. (Contributed by Scott Fenton, 17-May-2025.) |
| Ref | Expression |
|---|---|
| subsfo | ⊢ -s :( No × No )–onto→ No |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | subsf 28094 | . 2 ⊢ -s :( No × No )⟶ No | |
| 2 | 0sno 27871 | . . . . 5 ⊢ 0s ∈ No | |
| 3 | opelxpi 5722 | . . . . 5 ⊢ ((𝑥 ∈ No ∧ 0s ∈ No ) → 〈𝑥, 0s 〉 ∈ ( No × No )) | |
| 4 | 2, 3 | mpan2 691 | . . . 4 ⊢ (𝑥 ∈ No → 〈𝑥, 0s 〉 ∈ ( No × No )) |
| 5 | subsval 28090 | . . . . . 6 ⊢ ((𝑥 ∈ No ∧ 0s ∈ No ) → (𝑥 -s 0s ) = (𝑥 +s ( -us ‘ 0s ))) | |
| 6 | 2, 5 | mpan2 691 | . . . . 5 ⊢ (𝑥 ∈ No → (𝑥 -s 0s ) = (𝑥 +s ( -us ‘ 0s ))) |
| 7 | negs0s 28058 | . . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s | |
| 8 | 7 | oveq2i 7442 | . . . . . 6 ⊢ (𝑥 +s ( -us ‘ 0s )) = (𝑥 +s 0s ) |
| 9 | addsrid 27997 | . . . . . 6 ⊢ (𝑥 ∈ No → (𝑥 +s 0s ) = 𝑥) | |
| 10 | 8, 9 | eqtrid 2789 | . . . . 5 ⊢ (𝑥 ∈ No → (𝑥 +s ( -us ‘ 0s )) = 𝑥) |
| 11 | 6, 10 | eqtr2d 2778 | . . . 4 ⊢ (𝑥 ∈ No → 𝑥 = (𝑥 -s 0s )) |
| 12 | fveq2 6906 | . . . . . 6 ⊢ (𝑦 = 〈𝑥, 0s 〉 → ( -s ‘𝑦) = ( -s ‘〈𝑥, 0s 〉)) | |
| 13 | df-ov 7434 | . . . . . 6 ⊢ (𝑥 -s 0s ) = ( -s ‘〈𝑥, 0s 〉) | |
| 14 | 12, 13 | eqtr4di 2795 | . . . . 5 ⊢ (𝑦 = 〈𝑥, 0s 〉 → ( -s ‘𝑦) = (𝑥 -s 0s )) |
| 15 | 14 | rspceeqv 3645 | . . . 4 ⊢ ((〈𝑥, 0s 〉 ∈ ( No × No ) ∧ 𝑥 = (𝑥 -s 0s )) → ∃𝑦 ∈ ( No × No )𝑥 = ( -s ‘𝑦)) |
| 16 | 4, 11, 15 | syl2anc 584 | . . 3 ⊢ (𝑥 ∈ No → ∃𝑦 ∈ ( No × No )𝑥 = ( -s ‘𝑦)) |
| 17 | 16 | rgen 3063 | . 2 ⊢ ∀𝑥 ∈ No ∃𝑦 ∈ ( No × No )𝑥 = ( -s ‘𝑦) |
| 18 | dffo3 7122 | . 2 ⊢ ( -s :( No × No )–onto→ No ↔ ( -s :( No × No )⟶ No ∧ ∀𝑥 ∈ No ∃𝑦 ∈ ( No × No )𝑥 = ( -s ‘𝑦))) | |
| 19 | 1, 17, 18 | mpbir2an 711 | 1 ⊢ -s :( No × No )–onto→ No |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 〈cop 4632 × cxp 5683 ⟶wf 6557 –onto→wfo 6559 ‘cfv 6561 (class class class)co 7431 No csur 27684 0s c0s 27867 +s cadds 27992 -us cnegs 28051 -s csubs 28052 |
| 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| 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-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-int 4947 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-se 5638 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-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-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-1o 8506 df-2o 8507 df-nadd 8704 df-no 27687 df-slt 27688 df-bday 27689 df-sslt 27826 df-scut 27828 df-0s 27869 df-made 27886 df-old 27887 df-left 27889 df-right 27890 df-norec 27971 df-norec2 27982 df-adds 27993 df-negs 28053 df-subs 28054 |
| This theorem is referenced by: zssno 28367 |
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