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Theorem addsproplem1 27860
Description: Lemma for surreal addition properties. To prove closure on surreal addition we need to prove that addition is compatible with order at the same time. We do this by inducting over the maximum of two natural sums of the birthdays of surreals numbers. In the final step we will loop around and use tfr3 8411 to prove this of all surreals. This first lemma just instantiates the inductive hypothesis so we do not need to do it continuously throughout the proof. (Contributed by Scott Fenton, 21-Jan-2025.)
Hypotheses
Ref Expression
addsproplem.1 (𝜑 → ∀𝑥 No 𝑦 No 𝑧 No (((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
addsproplem1.2 (𝜑𝐴 No )
addsproplem1.3 (𝜑𝐵 No )
addsproplem1.4 (𝜑𝐶 No )
addsproplem1.5 (𝜑 → ((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝐶))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))))
Assertion
Ref Expression
addsproplem1 (𝜑 → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴))))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑦,𝐵,𝑧   𝑧,𝐶   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧   𝑥,𝑍,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem addsproplem1
StepHypRef Expression
1 addsproplem1.2 . . 3 (𝜑𝐴 No )
2 addsproplem1.3 . . 3 (𝜑𝐵 No )
3 addsproplem1.4 . . 3 (𝜑𝐶 No )
41, 2, 33jca 1126 . 2 (𝜑 → (𝐴 No 𝐵 No 𝐶 No ))
5 addsproplem.1 . 2 (𝜑 → ∀𝑥 No 𝑦 No 𝑧 No (((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
6 addsproplem1.5 . 2 (𝜑 → ((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝐶))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))))
7 fveq2 6891 . . . . . . 7 (𝑥 = 𝐴 → ( bday 𝑥) = ( bday 𝐴))
87oveq1d 7429 . . . . . 6 (𝑥 = 𝐴 → (( bday 𝑥) +no ( bday 𝑦)) = (( bday 𝐴) +no ( bday 𝑦)))
97oveq1d 7429 . . . . . 6 (𝑥 = 𝐴 → (( bday 𝑥) +no ( bday 𝑧)) = (( bday 𝐴) +no ( bday 𝑧)))
108, 9uneq12d 4160 . . . . 5 (𝑥 = 𝐴 → ((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) = ((( bday 𝐴) +no ( bday 𝑦)) ∪ (( bday 𝐴) +no ( bday 𝑧))))
1110eleq1d 2813 . . . 4 (𝑥 = 𝐴 → (((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) ↔ ((( bday 𝐴) +no ( bday 𝑦)) ∪ (( bday 𝐴) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍)))))
12 oveq1 7421 . . . . . 6 (𝑥 = 𝐴 → (𝑥 +s 𝑦) = (𝐴 +s 𝑦))
1312eleq1d 2813 . . . . 5 (𝑥 = 𝐴 → ((𝑥 +s 𝑦) ∈ No ↔ (𝐴 +s 𝑦) ∈ No ))
14 oveq2 7422 . . . . . . 7 (𝑥 = 𝐴 → (𝑦 +s 𝑥) = (𝑦 +s 𝐴))
15 oveq2 7422 . . . . . . 7 (𝑥 = 𝐴 → (𝑧 +s 𝑥) = (𝑧 +s 𝐴))
1614, 15breq12d 5155 . . . . . 6 (𝑥 = 𝐴 → ((𝑦 +s 𝑥) <s (𝑧 +s 𝑥) ↔ (𝑦 +s 𝐴) <s (𝑧 +s 𝐴)))
1716imbi2d 340 . . . . 5 (𝑥 = 𝐴 → ((𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)) ↔ (𝑦 <s 𝑧 → (𝑦 +s 𝐴) <s (𝑧 +s 𝐴))))
1813, 17anbi12d 630 . . . 4 (𝑥 = 𝐴 → (((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))) ↔ ((𝐴 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝐴) <s (𝑧 +s 𝐴)))))
1911, 18imbi12d 344 . . 3 (𝑥 = 𝐴 → ((((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))) ↔ (((( bday 𝐴) +no ( bday 𝑦)) ∪ (( bday 𝐴) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝐴 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝐴) <s (𝑧 +s 𝐴))))))
20 fveq2 6891 . . . . . . 7 (𝑦 = 𝐵 → ( bday 𝑦) = ( bday 𝐵))
2120oveq2d 7430 . . . . . 6 (𝑦 = 𝐵 → (( bday 𝐴) +no ( bday 𝑦)) = (( bday 𝐴) +no ( bday 𝐵)))
2221uneq1d 4158 . . . . 5 (𝑦 = 𝐵 → ((( bday 𝐴) +no ( bday 𝑦)) ∪ (( bday 𝐴) +no ( bday 𝑧))) = ((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝑧))))
2322eleq1d 2813 . . . 4 (𝑦 = 𝐵 → (((( bday 𝐴) +no ( bday 𝑦)) ∪ (( bday 𝐴) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) ↔ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍)))))
24 oveq2 7422 . . . . . 6 (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
2524eleq1d 2813 . . . . 5 (𝑦 = 𝐵 → ((𝐴 +s 𝑦) ∈ No ↔ (𝐴 +s 𝐵) ∈ No ))
26 breq1 5145 . . . . . 6 (𝑦 = 𝐵 → (𝑦 <s 𝑧𝐵 <s 𝑧))
27 oveq1 7421 . . . . . . 7 (𝑦 = 𝐵 → (𝑦 +s 𝐴) = (𝐵 +s 𝐴))
2827breq1d 5152 . . . . . 6 (𝑦 = 𝐵 → ((𝑦 +s 𝐴) <s (𝑧 +s 𝐴) ↔ (𝐵 +s 𝐴) <s (𝑧 +s 𝐴)))
2926, 28imbi12d 344 . . . . 5 (𝑦 = 𝐵 → ((𝑦 <s 𝑧 → (𝑦 +s 𝐴) <s (𝑧 +s 𝐴)) ↔ (𝐵 <s 𝑧 → (𝐵 +s 𝐴) <s (𝑧 +s 𝐴))))
3025, 29anbi12d 630 . . . 4 (𝑦 = 𝐵 → (((𝐴 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝐴) <s (𝑧 +s 𝐴))) ↔ ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝑧 → (𝐵 +s 𝐴) <s (𝑧 +s 𝐴)))))
3123, 30imbi12d 344 . . 3 (𝑦 = 𝐵 → ((((( bday 𝐴) +no ( bday 𝑦)) ∪ (( bday 𝐴) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝐴 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝐴) <s (𝑧 +s 𝐴)))) ↔ (((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝑧 → (𝐵 +s 𝐴) <s (𝑧 +s 𝐴))))))
32 fveq2 6891 . . . . . . 7 (𝑧 = 𝐶 → ( bday 𝑧) = ( bday 𝐶))
3332oveq2d 7430 . . . . . 6 (𝑧 = 𝐶 → (( bday 𝐴) +no ( bday 𝑧)) = (( bday 𝐴) +no ( bday 𝐶)))
3433uneq2d 4159 . . . . 5 (𝑧 = 𝐶 → ((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝑧))) = ((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝐶))))
3534eleq1d 2813 . . . 4 (𝑧 = 𝐶 → (((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) ↔ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝐶))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍)))))
36 breq2 5146 . . . . . 6 (𝑧 = 𝐶 → (𝐵 <s 𝑧𝐵 <s 𝐶))
37 oveq1 7421 . . . . . . 7 (𝑧 = 𝐶 → (𝑧 +s 𝐴) = (𝐶 +s 𝐴))
3837breq2d 5154 . . . . . 6 (𝑧 = 𝐶 → ((𝐵 +s 𝐴) <s (𝑧 +s 𝐴) ↔ (𝐵 +s 𝐴) <s (𝐶 +s 𝐴)))
3936, 38imbi12d 344 . . . . 5 (𝑧 = 𝐶 → ((𝐵 <s 𝑧 → (𝐵 +s 𝐴) <s (𝑧 +s 𝐴)) ↔ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴))))
4039anbi2d 628 . . . 4 (𝑧 = 𝐶 → (((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝑧 → (𝐵 +s 𝐴) <s (𝑧 +s 𝐴))) ↔ ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴)))))
4135, 40imbi12d 344 . . 3 (𝑧 = 𝐶 → ((((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝑧 → (𝐵 +s 𝐴) <s (𝑧 +s 𝐴)))) ↔ (((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝐶))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴))))))
4219, 31, 41rspc3v 3623 . 2 ((𝐴 No 𝐵 No 𝐶 No ) → (∀𝑥 No 𝑦 No 𝑧 No (((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))) → (((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝐶))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴))))))
434, 5, 6, 42syl3c 66 1 (𝜑 → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1534  wcel 2099  wral 3056  cun 3942   class class class wbr 5142  cfv 6542  (class class class)co 7414   +no cnadd 8677   No csur 27547   <s cslt 27548   bday cbday 27549   +s cadds 27850
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-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-iota 6494  df-fv 6550  df-ov 7417
This theorem is referenced by:  addsproplem2  27861  addsproplem6  27865
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