Theorem addsproplem7 28391

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		bdayon 28168	. . . 4 ⊢ (bday‘𝑌) ∈ On
2		fvex 6921	. . . . 5 ⊢ (bday‘𝑌) ∈ V
3	2	elon 6389	. . . 4 ⊢ ((bday‘𝑌) ∈ On ↔ Ord (bday‘𝑌))
4	1, 3	mpbi 233	. . 3 ⊢ Ord (bday‘𝑌)
5		bdayon 28168	. . . 4 ⊢ (bday‘𝑍) ∈ On
6		fvex 6921	. . . . 5 ⊢ (bday‘𝑍) ∈ V
7	6	elon 6389	. . . 4 ⊢ ((bday‘𝑍) ∈ On ↔ Ord (bday‘𝑍))
8	5, 7	mpbi 233	. . 3 ⊢ Ord (bday‘𝑍)
9		ordtri3or 6413	. . 3 ⊢ ((Ord (bday‘𝑌) ∧ Ord (bday‘𝑍)) → ((bday‘𝑌) ∈ (bday‘𝑍) ∨ (bday‘𝑌) = (bday‘𝑍) ∨ (bday‘𝑍) ∈ (bday‘𝑌)))
10	4, 8, 9	mp2an 705	. 2 ⊢ ((bday‘𝑌) ∈ (bday‘𝑍) ∨ (bday‘𝑌) = (bday‘𝑍) ∨ (bday‘𝑍) ∈ (bday‘𝑌))
11		simpl 488	. . . . . 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 18	. . . . 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 18	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑌) ∈ (bday‘𝑍)) → 𝑋 ∈ No)
16		addspropord.3	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ No)
17	11, 16	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑌) ∈ (bday‘𝑍)) → 𝑌 ∈ No)
18		addspropord.4	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ No)
19	11, 18	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑌) ∈ (bday‘𝑍)) → 𝑍 ∈ No)
20		addspropord.5	. . . . . 6 ⊢ (𝜑 → 𝑌 <s 𝑍)
21	11, 20	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑌) ∈ (bday‘𝑍)) → 𝑌 <s 𝑍)
22		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑌) ∈ (bday‘𝑍)) → (bday‘𝑌) ∈ (bday‘𝑍))
23	13, 15, 17, 19, 21, 22	addsproplem4 28388	. . . 4 ⊢ ((𝜑 ∧ (bday‘𝑌) ∈ (bday‘𝑍)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
24	23	ex 418	. . 3 ⊢ (𝜑 → ((bday‘𝑌) ∈ (bday‘𝑍) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
25		simpl 488	. . . . . 6 ⊢ ((𝜑 ∧ (bday‘𝑌) = (bday‘𝑍)) → 𝜑)
26	25, 12	syl 18	. . . . 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 18	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑌) = (bday‘𝑍)) → 𝑋 ∈ No)
28	25, 16	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑌) = (bday‘𝑍)) → 𝑌 ∈ No)
29	25, 18	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑌) = (bday‘𝑍)) → 𝑍 ∈ No)
30	25, 20	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑌) = (bday‘𝑍)) → 𝑌 <s 𝑍)
31		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑌) = (bday‘𝑍)) → (bday‘𝑌) = (bday‘𝑍))
32	26, 27, 28, 29, 30, 31	addsproplem6 28390	. . . 4 ⊢ ((𝜑 ∧ (bday‘𝑌) = (bday‘𝑍)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
33	32	ex 418	. . 3 ⊢ (𝜑 → ((bday‘𝑌) = (bday‘𝑍) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
34	12	adantr 486	. . . . 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 486	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑍) ∈ (bday‘𝑌)) → 𝑋 ∈ No)
36	16	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑍) ∈ (bday‘𝑌)) → 𝑌 ∈ No)
37	18	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑍) ∈ (bday‘𝑌)) → 𝑍 ∈ No)
38	20	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑍) ∈ (bday‘𝑌)) → 𝑌 <s 𝑍)
39		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝑍) ∈ (bday‘𝑌)) → (bday‘𝑍) ∈ (bday‘𝑌))
40	34, 35, 36, 37, 38, 39	addsproplem5 28389	. . . 4 ⊢ ((𝜑 ∧ (bday‘𝑍) ∈ (bday‘𝑌)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
41	40	ex 418	. . 3 ⊢ (𝜑 → ((bday‘𝑍) ∈ (bday‘𝑌) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
42	24, 33, 41	3jaod 1457	. 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 21	1 ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ w3o 1102   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ∪ cun 4152   class class class wbr 5106  Ord word 6379  Oncon0 6380  ‘cfv 6558  (class class class)co 7443   +no cnadd 8702  Nocsur 28027   <s clts 28028  bdaycbday 28029   +_s cadds 28375

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-1o 8504  df-2o 8505  df-nadd 8703  df-no 28030  df-lts 28031  df-bday 28032  df-slts 28174  df-cuts 28176  df-0s 28223  df-made 28243  df-old 28244  df-left 28246  df-right 28247  df-norec2 28365  df-adds 28376

This theorem is used by:  addsprop  28392