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Mirrors > Home > MPE Home > Th. List > addsfo | Structured version Visualization version GIF version |
Description: Surreal addition is onto. (Contributed by Scott Fenton, 21-Jan-2025.) |
Ref | Expression |
---|---|
addsfo | ⊢ +s :( No × No )–onto→ No |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addsf 27295 | . 2 ⊢ +s :( No × No )⟶ No | |
2 | 0sno 27168 | . . . . 5 ⊢ 0s ∈ No | |
3 | opelxpi 5671 | . . . . 5 ⊢ ((𝑧 ∈ No ∧ 0s ∈ No ) → ⟨𝑧, 0s ⟩ ∈ ( No × No )) | |
4 | 2, 3 | mpan2 690 | . . . 4 ⊢ (𝑧 ∈ No → ⟨𝑧, 0s ⟩ ∈ ( No × No )) |
5 | addsid1 27279 | . . . . 5 ⊢ (𝑧 ∈ No → (𝑧 +s 0s ) = 𝑧) | |
6 | 5 | eqcomd 2743 | . . . 4 ⊢ (𝑧 ∈ No → 𝑧 = (𝑧 +s 0s )) |
7 | fveq2 6843 | . . . . . 6 ⊢ (𝑥 = ⟨𝑧, 0s ⟩ → ( +s ‘𝑥) = ( +s ‘⟨𝑧, 0s ⟩)) | |
8 | df-ov 7361 | . . . . . 6 ⊢ (𝑧 +s 0s ) = ( +s ‘⟨𝑧, 0s ⟩) | |
9 | 7, 8 | eqtr4di 2795 | . . . . 5 ⊢ (𝑥 = ⟨𝑧, 0s ⟩ → ( +s ‘𝑥) = (𝑧 +s 0s )) |
10 | 9 | rspceeqv 3596 | . . . 4 ⊢ ((⟨𝑧, 0s ⟩ ∈ ( No × No ) ∧ 𝑧 = (𝑧 +s 0s )) → ∃𝑥 ∈ ( No × No )𝑧 = ( +s ‘𝑥)) |
11 | 4, 6, 10 | syl2anc 585 | . . 3 ⊢ (𝑧 ∈ No → ∃𝑥 ∈ ( No × No )𝑧 = ( +s ‘𝑥)) |
12 | 11 | rgen 3067 | . 2 ⊢ ∀𝑧 ∈ No ∃𝑥 ∈ ( No × No )𝑧 = ( +s ‘𝑥) |
13 | dffo3 7053 | . 2 ⊢ ( +s :( No × No )–onto→ No ↔ ( +s :( No × No )⟶ No ∧ ∀𝑧 ∈ No ∃𝑥 ∈ ( No × No )𝑧 = ( +s ‘𝑥))) | |
14 | 1, 12, 13 | mpbir2an 710 | 1 ⊢ +s :( No × No )–onto→ No |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 ∀wral 3065 ∃wrex 3074 ⟨cop 4593 × cxp 5632 ⟶wf 6493 –onto→wfo 6495 ‘cfv 6497 (class class class)co 7358 No csur 26991 0s c0s 27164 +s cadds 27274 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-1o 8413 df-2o 8414 df-nadd 8613 df-no 26994 df-slt 26995 df-bday 26996 df-sslt 27124 df-scut 27126 df-0s 27166 df-made 27180 df-old 27181 df-left 27183 df-right 27184 df-norec2 27264 df-adds 27275 |
This theorem is referenced by: (None) |
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