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Theorem addsproplem1 28002
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 8439 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 1129 . 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 6906 . . . . . . 7 (𝑥 = 𝐴 → ( bday 𝑥) = ( bday 𝐴))
87oveq1d 7446 . . . . . 6 (𝑥 = 𝐴 → (( bday 𝑥) +no ( bday 𝑦)) = (( bday 𝐴) +no ( bday 𝑦)))
97oveq1d 7446 . . . . . 6 (𝑥 = 𝐴 → (( bday 𝑥) +no ( bday 𝑧)) = (( bday 𝐴) +no ( bday 𝑧)))
108, 9uneq12d 4169 . . . . 5 (𝑥 = 𝐴 → ((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) = ((( bday 𝐴) +no ( bday 𝑦)) ∪ (( bday 𝐴) +no ( bday 𝑧))))
1110eleq1d 2826 . . . 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 7438 . . . . . 6 (𝑥 = 𝐴 → (𝑥 +s 𝑦) = (𝐴 +s 𝑦))
1312eleq1d 2826 . . . . 5 (𝑥 = 𝐴 → ((𝑥 +s 𝑦) ∈ No ↔ (𝐴 +s 𝑦) ∈ No ))
14 oveq2 7439 . . . . . . 7 (𝑥 = 𝐴 → (𝑦 +s 𝑥) = (𝑦 +s 𝐴))
15 oveq2 7439 . . . . . . 7 (𝑥 = 𝐴 → (𝑧 +s 𝑥) = (𝑧 +s 𝐴))
1614, 15breq12d 5156 . . . . . 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 632 . . . 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 6906 . . . . . . 7 (𝑦 = 𝐵 → ( bday 𝑦) = ( bday 𝐵))
2120oveq2d 7447 . . . . . 6 (𝑦 = 𝐵 → (( bday 𝐴) +no ( bday 𝑦)) = (( bday 𝐴) +no ( bday 𝐵)))
2221uneq1d 4167 . . . . 5 (𝑦 = 𝐵 → ((( bday 𝐴) +no ( bday 𝑦)) ∪ (( bday 𝐴) +no ( bday 𝑧))) = ((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝑧))))
2322eleq1d 2826 . . . 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 7439 . . . . . 6 (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
2524eleq1d 2826 . . . . 5 (𝑦 = 𝐵 → ((𝐴 +s 𝑦) ∈ No ↔ (𝐴 +s 𝐵) ∈ No ))
26 breq1 5146 . . . . . 6 (𝑦 = 𝐵 → (𝑦 <s 𝑧𝐵 <s 𝑧))
27 oveq1 7438 . . . . . . 7 (𝑦 = 𝐵 → (𝑦 +s 𝐴) = (𝐵 +s 𝐴))
2827breq1d 5153 . . . . . 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 632 . . . 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 6906 . . . . . . 7 (𝑧 = 𝐶 → ( bday 𝑧) = ( bday 𝐶))
3332oveq2d 7447 . . . . . 6 (𝑧 = 𝐶 → (( bday 𝐴) +no ( bday 𝑧)) = (( bday 𝐴) +no ( bday 𝐶)))
3433uneq2d 4168 . . . . 5 (𝑧 = 𝐶 → ((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝑧))) = ((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝐶))))
3534eleq1d 2826 . . . 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 5147 . . . . . 6 (𝑧 = 𝐶 → (𝐵 <s 𝑧𝐵 <s 𝐶))
37 oveq1 7438 . . . . . . 7 (𝑧 = 𝐶 → (𝑧 +s 𝐴) = (𝐶 +s 𝐴))
3837breq2d 5155 . . . . . 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 630 . . . 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 3638 . 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 1087   = wceq 1540  wcel 2108  wral 3061  cun 3949   class class class wbr 5143  cfv 6561  (class class class)co 7431   +no cnadd 8703   No csur 27684   <s cslt 27685   bday cbday 27686   +s cadds 27992
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-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434
This theorem is referenced by:  addsproplem2  28003  addsproplem6  28007
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