Theorem mulsproplem3 28561

Description: Lemma for surreal multiplication. Under the inductive hypothesis, the product of 𝐴 itself and a member of the old set of 𝐵 is a surreal number. (Contributed by Scott Fenton, 4-Mar-2025.)

Hypotheses

Ref	Expression
mulsproplem.1	⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No ((((bday‘𝑎) +no (bday‘𝑏)) ∪ ((((bday‘𝑐) +no (bday‘𝑒)) ∪ ((bday‘𝑑) +no (bday‘𝑓))) ∪ (((bday‘𝑐) +no (bday‘𝑓)) ∪ ((bday‘𝑑) +no (bday‘𝑒))))) ∈ (((bday‘𝐴) +no (bday‘𝐵)) ∪ ((((bday‘𝐶) +no (bday‘𝐸)) ∪ ((bday‘𝐷) +no (bday‘𝐹))) ∪ (((bday‘𝐶) +no (bday‘𝐹)) ∪ ((bday‘𝐷) +no (bday‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) −s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) −s (𝑑 ·s 𝑒))))))
mulsproplem3.1	⊢ (𝜑 → 𝐴 ∈ No)
mulsproplem3.2	⊢ (𝜑 → 𝑌 ∈ (O‘(bday‘𝐵)))

Assertion

Ref	Expression
mulsproplem3	⊢ (𝜑 → (𝐴 ·s 𝑌) ∈ No)

Distinct variable groups:   𝐴,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐵,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐶,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐷,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐸,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐹,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝑌,𝑏,𝑐,𝑑,𝑒,𝑓

Allowed substitution hints:   𝜑(𝑒, 𝑓, 𝑎, 𝑏, 𝑐, 𝑑)   𝑌(𝑎)

Proof of Theorem mulsproplem3

Step	Hyp	Ref	Expression
1		mulsproplem.1	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No ((((bday‘𝑎) +no (bday‘𝑏)) ∪ ((((bday‘𝑐) +no (bday‘𝑒)) ∪ ((bday‘𝑑) +no (bday‘𝑓))) ∪ (((bday‘𝑐) +no (bday‘𝑓)) ∪ ((bday‘𝑑) +no (bday‘𝑒))))) ∈ (((bday‘𝐴) +no (bday‘𝐵)) ∪ ((((bday‘𝐶) +no (bday‘𝐸)) ∪ ((bday‘𝐷) +no (bday‘𝐹))) ∪ (((bday‘𝐶) +no (bday‘𝐹)) ∪ ((bday‘𝐷) +no (bday‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) −s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) −s (𝑑 ·s 𝑒))))))
2		mulsproplem3.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ No)
3		mulsproplem3.2	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (O‘(bday‘𝐵)))
4	3	oldnod 28290	. . 3 ⊢ (𝜑 → 𝑌 ∈ No)
5		0no 28252	. . . 4 ⊢ 0s ∈ No
6	5	a1i 11	. . 3 ⊢ (𝜑 → 0s ∈ No)
7		bday0 28254	. . . . . . . . . . . 12 ⊢ (bday‘ 0s ) = ∅
8	7, 7	oveq12i 7463	. . . . . . . . . . 11 ⊢ ((bday‘ 0s ) +no (bday‘ 0s )) = (∅ +no ∅)
9		0elon 6444	. . . . . . . . . . . 12 ⊢ ∅ ∈ On
10		naddrid 8731	. . . . . . . . . . . 12 ⊢ (∅ ∈ On → (∅ +no ∅) = ∅)
11	9, 10	ax-mp 5	. . . . . . . . . . 11 ⊢ (∅ +no ∅) = ∅
12	8, 11	eqtri 2786	. . . . . . . . . 10 ⊢ ((bday‘ 0s ) +no (bday‘ 0s )) = ∅
13	12, 12	uneq12i 4168	. . . . . . . . 9 ⊢ (((bday‘ 0s ) +no (bday‘ 0s )) ∪ ((bday‘ 0s ) +no (bday‘ 0s ))) = (∅ ∪ ∅)
14		un0 4347	. . . . . . . . 9 ⊢ (∅ ∪ ∅) = ∅
15	13, 14	eqtri 2786	. . . . . . . 8 ⊢ (((bday‘ 0s ) +no (bday‘ 0s )) ∪ ((bday‘ 0s ) +no (bday‘ 0s ))) = ∅
16	15, 15	uneq12i 4168	. . . . . . 7 ⊢ ((((bday‘ 0s ) +no (bday‘ 0s )) ∪ ((bday‘ 0s ) +no (bday‘ 0s ))) ∪ (((bday‘ 0s ) +no (bday‘ 0s )) ∪ ((bday‘ 0s ) +no (bday‘ 0s )))) = (∅ ∪ ∅)
17	16, 14	eqtri 2786	. . . . . 6 ⊢ ((((bday‘ 0s ) +no (bday‘ 0s )) ∪ ((bday‘ 0s ) +no (bday‘ 0s ))) ∪ (((bday‘ 0s ) +no (bday‘ 0s )) ∪ ((bday‘ 0s ) +no (bday‘ 0s )))) = ∅
18	17	uneq2i 4167	. . . . 5 ⊢ (((bday‘𝐴) +no (bday‘𝑌)) ∪ ((((bday‘ 0s ) +no (bday‘ 0s )) ∪ ((bday‘ 0s ) +no (bday‘ 0s ))) ∪ (((bday‘ 0s ) +no (bday‘ 0s )) ∪ ((bday‘ 0s ) +no (bday‘ 0s ))))) = (((bday‘𝐴) +no (bday‘𝑌)) ∪ ∅)
19		un0 4347	. . . . 5 ⊢ (((bday‘𝐴) +no (bday‘𝑌)) ∪ ∅) = ((bday‘𝐴) +no (bday‘𝑌))
20	18, 19	eqtri 2786	. . . 4 ⊢ (((bday‘𝐴) +no (bday‘𝑌)) ∪ ((((bday‘ 0s ) +no (bday‘ 0s )) ∪ ((bday‘ 0s ) +no (bday‘ 0s ))) ∪ (((bday‘ 0s ) +no (bday‘ 0s )) ∪ ((bday‘ 0s ) +no (bday‘ 0s ))))) = ((bday‘𝐴) +no (bday‘𝑌))
21		oldbdayim 28332	. . . . . . 7 ⊢ (𝑌 ∈ (O‘(bday‘𝐵)) → (bday‘𝑌) ∈ (bday‘𝐵))
22	3, 21	syl 18	. . . . . 6 ⊢ (𝜑 → (bday‘𝑌) ∈ (bday‘𝐵))
23		bdayon 28195	. . . . . . 7 ⊢ (bday‘𝑌) ∈ On
24		bdayon 28195	. . . . . . 7 ⊢ (bday‘𝐵) ∈ On
25		bdayon 28195	. . . . . . 7 ⊢ (bday‘𝐴) ∈ On
26		naddel2 8736	. . . . . . 7 ⊢ (((bday‘𝑌) ∈ On ∧ (bday‘𝐵) ∈ On ∧ (bday‘𝐴) ∈ On) → ((bday‘𝑌) ∈ (bday‘𝐵) ↔ ((bday‘𝐴) +no (bday‘𝑌)) ∈ ((bday‘𝐴) +no (bday‘𝐵))))
27	23, 24, 25, 26	mp3an 1491	. . . . . 6 ⊢ ((bday‘𝑌) ∈ (bday‘𝐵) ↔ ((bday‘𝐴) +no (bday‘𝑌)) ∈ ((bday‘𝐴) +no (bday‘𝐵)))
28	22, 27	sylib 221	. . . . 5 ⊢ (𝜑 → ((bday‘𝐴) +no (bday‘𝑌)) ∈ ((bday‘𝐴) +no (bday‘𝐵)))
29		elun1 4183	. . . . 5 ⊢ (((bday‘𝐴) +no (bday‘𝑌)) ∈ ((bday‘𝐴) +no (bday‘𝐵)) → ((bday‘𝐴) +no (bday‘𝑌)) ∈ (((bday‘𝐴) +no (bday‘𝐵)) ∪ ((((bday‘𝐶) +no (bday‘𝐸)) ∪ ((bday‘𝐷) +no (bday‘𝐹))) ∪ (((bday‘𝐶) +no (bday‘𝐹)) ∪ ((bday‘𝐷) +no (bday‘𝐸))))))
30	28, 29	syl 18	. . . 4 ⊢ (𝜑 → ((bday‘𝐴) +no (bday‘𝑌)) ∈ (((bday‘𝐴) +no (bday‘𝐵)) ∪ ((((bday‘𝐶) +no (bday‘𝐸)) ∪ ((bday‘𝐷) +no (bday‘𝐹))) ∪ (((bday‘𝐶) +no (bday‘𝐹)) ∪ ((bday‘𝐷) +no (bday‘𝐸))))))
31	20, 30	eqeltrid 2868	. . 3 ⊢ (𝜑 → (((bday‘𝐴) +no (bday‘𝑌)) ∪ ((((bday‘ 0s ) +no (bday‘ 0s )) ∪ ((bday‘ 0s ) +no (bday‘ 0s ))) ∪ (((bday‘ 0s ) +no (bday‘ 0s )) ∪ ((bday‘ 0s ) +no (bday‘ 0s ))))) ∈ (((bday‘𝐴) +no (bday‘𝐵)) ∪ ((((bday‘𝐶) +no (bday‘𝐸)) ∪ ((bday‘𝐷) +no (bday‘𝐹))) ∪ (((bday‘𝐶) +no (bday‘𝐹)) ∪ ((bday‘𝐷) +no (bday‘𝐸))))))
32	1, 2, 4, 6, 6, 6, 6, 31	mulsproplem1 28559	. 2 ⊢ (𝜑 → ((𝐴 ·s 𝑌) ∈ No ∧ (( 0s <s 0s ∧ 0s <s 0s ) → (( 0s ·s 0s ) −s ( 0s ·s 0s )) <s (( 0s ·s 0s ) −s ( 0s ·s 0s )))))
33	32	simpld 500	1 ⊢ (𝜑 → (𝐴 ·s 𝑌) ∈ No)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ∪ cun 4152  ∅c0 4281   class class class wbr 5108  Oncon0 6388  ‘cfv 6566  (class class class)co 7451   +no cnadd 8712  Nocsur 28054   <s clts 28055  bdaycbday 28056   0_s c0s 28248  Ocold 28266   −_s csubs 28463   ·_s cmuls 28549

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 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-1o 8514  df-2o 8515  df-nadd 8713  df-no 28057  df-lts 28058  df-bday 28059  df-slts 28201  df-cuts 28203  df-0s 28250  df-made 28270  df-old 28271

This theorem is used by:  mulsproplem5  28563  mulsproplem6  28564  mulsproplem7  28565  mulsproplem8  28566  mulsproplem9  28567  mulsproplem14  28572