Theorem addsproplem7 27448

Description: Lemma for surreal addition properties. Putting together the three previous lemmas, we now show the second half of the inductive hypothesis unconditionally. (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)))))
addspropord.2	⊢ (𝜑 → 𝑋 ∈ No )
addspropord.3	⊢ (𝜑 → 𝑌 ∈ No )
addspropord.4	⊢ (𝜑 → 𝑍 ∈ No )
addspropord.5	⊢ (𝜑 → 𝑌 <s 𝑍)

Assertion

Ref	Expression
addsproplem7	⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))

Distinct variable groups:   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧   𝑥,𝑍,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem addsproplem7

Step	Hyp	Ref	Expression
1		bdayelon 27267	. . . 4 ⊢ ( bday ‘𝑌) ∈ On
2		fvex 6901	. . . . 5 ⊢ ( bday ‘𝑌) ∈ V
3	2	elon 6370	. . . 4 ⊢ (( bday ‘𝑌) ∈ On ↔ Ord ( bday ‘𝑌))
4	1, 3	mpbi 229	. . 3 ⊢ Ord ( bday ‘𝑌)
5		bdayelon 27267	. . . 4 ⊢ ( bday ‘𝑍) ∈ On
6		fvex 6901	. . . . 5 ⊢ ( bday ‘𝑍) ∈ V
7	6	elon 6370	. . . 4 ⊢ (( bday ‘𝑍) ∈ On ↔ Ord ( bday ‘𝑍))
8	5, 7	mpbi 229	. . 3 ⊢ Ord ( bday ‘𝑍)
9		ordtri3or 6393	. . 3 ⊢ ((Ord ( bday ‘𝑌) ∧ Ord ( bday ‘𝑍)) → (( bday ‘𝑌) ∈ ( bday ‘𝑍) ∨ ( bday ‘𝑌) = ( bday ‘𝑍) ∨ ( bday ‘𝑍) ∈ ( bday ‘𝑌)))
10	4, 8, 9	mp2an 690	. 2 ⊢ (( bday ‘𝑌) ∈ ( bday ‘𝑍) ∨ ( bday ‘𝑌) = ( bday ‘𝑍) ∨ ( bday ‘𝑍) ∈ ( bday ‘𝑌))
11		simpl 483	. . . . . 6 ⊢ ((𝜑 ∧ ( bday ‘𝑌) ∈ ( bday ‘𝑍)) → 𝜑)
12		addsproplem.1	. . . . . 6 ⊢ (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)))))
13	11, 12	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) ∈ ( bday ‘𝑍)) → ∀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)))))
14		addspropord.2	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ No )
15	11, 14	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) ∈ ( bday ‘𝑍)) → 𝑋 ∈ No )
16		addspropord.3	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ No )
17	11, 16	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) ∈ ( bday ‘𝑍)) → 𝑌 ∈ No )
18		addspropord.4	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ No )
19	11, 18	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) ∈ ( bday ‘𝑍)) → 𝑍 ∈ No )
20		addspropord.5	. . . . . 6 ⊢ (𝜑 → 𝑌 <s 𝑍)
21	11, 20	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) ∈ ( bday ‘𝑍)) → 𝑌 <s 𝑍)
22		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) ∈ ( bday ‘𝑍)) → ( bday ‘𝑌) ∈ ( bday ‘𝑍))
23	13, 15, 17, 19, 21, 22	addsproplem4 27445	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝑌) ∈ ( bday ‘𝑍)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
24	23	ex 413	. . 3 ⊢ (𝜑 → (( bday ‘𝑌) ∈ ( bday ‘𝑍) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
25		simpl 483	. . . . . 6 ⊢ ((𝜑 ∧ ( bday ‘𝑌) = ( bday ‘𝑍)) → 𝜑)
26	25, 12	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) = ( bday ‘𝑍)) → ∀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)))))
27	25, 14	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) = ( bday ‘𝑍)) → 𝑋 ∈ No )
28	25, 16	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) = ( bday ‘𝑍)) → 𝑌 ∈ No )
29	25, 18	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) = ( bday ‘𝑍)) → 𝑍 ∈ No )
30	25, 20	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) = ( bday ‘𝑍)) → 𝑌 <s 𝑍)
31		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) = ( bday ‘𝑍)) → ( bday ‘𝑌) = ( bday ‘𝑍))
32	26, 27, 28, 29, 30, 31	addsproplem6 27447	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝑌) = ( bday ‘𝑍)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
33	32	ex 413	. . 3 ⊢ (𝜑 → (( bday ‘𝑌) = ( bday ‘𝑍) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
34	12	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → ∀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)))))
35	14	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → 𝑋 ∈ No )
36	16	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → 𝑌 ∈ No )
37	18	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → 𝑍 ∈ No )
38	20	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → 𝑌 <s 𝑍)
39		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → ( bday ‘𝑍) ∈ ( bday ‘𝑌))
40	34, 35, 36, 37, 38, 39	addsproplem5 27446	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
41	40	ex 413	. . 3 ⊢ (𝜑 → (( bday ‘𝑍) ∈ ( bday ‘𝑌) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
42	24, 33, 41	3jaod 1428	. 2 ⊢ (𝜑 → ((( bday ‘𝑌) ∈ ( bday ‘𝑍) ∨ ( bday ‘𝑌) = ( bday ‘𝑍) ∨ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
43	10, 42	mpi 20	1 ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))

Syntax hints:   → wi 4   ∧ wa 396   ∨ w3o 1086   = wceq 1541   ∈ wcel 2106  ∀wral 3061   ∪ cun 3945   class class class wbr 5147  Ord word 6360  Oncon0 6361  ‘cfv 6540  (class class class)co 7405   +no cnadd 8660   No csur 27132   <s cslt 27133   bday cbday 27134   +_s cadds 27432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-1o 8462  df-2o 8463  df-nadd 8661  df-no 27135  df-slt 27136  df-bday 27137  df-sslt 27272  df-scut 27274  df-0s 27314  df-made 27331  df-old 27332  df-left 27334  df-right 27335  df-norec2 27422  df-adds 27433

This theorem is referenced by:  addsprop  27449